P H Y S I C S U N I O N M A T H E M A T I C S. Physics II. Kinematics. Supported by the National Science Foundation (DRL ).

Transcription

1 P H Y S I C S U N I O N M A T H E M A T I C S Physics II Kinematics Supported by the National Science Foundation (DRL ).

2 PUM Physics II Kinematics Most of the module activities were adapted from: A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, Used with permission. Some activities area based on FMCE (Thornton and Sokoloff), on Ranking Task Exercises in Physics (O Kuma, Maloney, and Hieggelke), and on the work of D. Schwartz. Contributions of: E. Etkina, T. Bartiromo, S. Brahmia, J. Chia, C. D Amato, J. Flakker, J. Finley, H. Lopez, R. Newman, J. Santonacita, E. Siebenmann, R. Therkorn, K. Thomas, M. Trinh. This material is based upon work supported by the National Science Foundation under Grant DRL Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF). 2 PUM Kinematics

3 Table of Contents LESSON 1: ESSENTIALS TO DESCRIBING MOTION 4 LESSON 2: DESCRIBING MOTION 9 LESSON 3: REASONING IN KINEMATICS: INVENTING AN INDEX 14 LESSON 3A: HOW FAST ARE YOU MOVING? 19 LESSON 4: THE MOVING MAN 26 LESSON 5: HOW MANY VELOCITIES CAN AN OBJECT HAVE? 32 LESSON 6: USING THE LANGUAGE OF MATHEMATICS WHILE DOING PHYSICS 37 LESSON 7: USING VELOCITY-VERSUS-TIME GRAPHS TO FIND DISPLACEMENT 41 LESSON 8: LAB: WILL THE CARS EVER MEET? 46 LESSON 9: MOTION DIAGRAMS 52 LESSON 10: FREE FALLING? 57 LESSON 10B*: THE TRUTH BEHIND GRAPHIC REPRESENTATIONS 58 LESSON 11: MOTION OF A FALLING OBJECT 60 LESSON 12: POSITION OF AN ACCELERATING OBJECT AS A FUNCTION OF TIME 71 LESSON 13: EXPERIMENTAL DESIGN 77 LESSON 14: DETAILS OF THE THROW 79 LESSON 15: PUTTING IT ALL TOGETHER 82 LESSON 16: PRACTICE 8889 PUM Kinematics 3

4 Lesson 1: Essentials to Describing Motion 1.1 Observe and Find a Pattern In this activity, a tube telescope is used to follow an object. During the experiments, the observer always keeps the object in sight through the telescope. Consider the following situations: a) In the first experiment, the teacher holds an object and is standing still. You are the observer; take note of the initial direction that the telescope points when you see the object through it. This is the original orientation of the telescope. Next, the teacher travels from right to left. - What happens to your telescope as you follow the object? - Draw a rough sketch to show what happens to the orientation of the telescope. b) For the next experiment, carefully observe your teacher. The teacher holds the ball the same way. This time, the teacher is the observer and looks at the object through the telescope while traveling from right to left. - What happens to the orientation of the teacher s telescope during the experiment? c) Based on your experiences in these two observational experiments: - Can you say whether the ball was moving during the two experiments? Explain your answer. (Hint: Compare what the different observers had to do to keep seeing the ball through their telescopes.) - In general, how do you know whether something is moving or not? - Name two observers who see you moving right now. How do you know? - Name two observers who see you stationary right now. How do you know? d) What if. - Chris watches a car through his scope. He is turning his scope to the left. In what direction is the car moving? - Jodi watches the same car but she is not moving the scope. How can this be? What might happen to the size of the car as Jodi watches it? 1.2 Observe and Find a Pattern Consider the following situation. A blue car moves along a street with two passengers. One sits in the front passenger seat of the car and the other passenger sits in the back seat. 4 PUM Kinematics Lesson 1: Essentials to Describing Motion

5 A red car moves in the same direction and is passing the blue car. A green car moving faster than the blue car, is directly behind the blue car There is a sidewalk along the road the cars are traveling and a pedestrian is standing on the sidewalk. Choose four students from your class to play the roles of the four people and recreate the situation. Describe the movement of the front passenger in the blue car as seen by each of the following observers: Observer Describe what she/he sees The person sitting in the backseat of the blue car. The pedestrian standing on the sidewalk as the blue car passes. The driver of the red car moving in the same direction and passing the blue car. A passenger in the green car Review your descriptions and answer the questions that follow. a) Imagine you are the backseat passenger in the blue car, how would you observe the other four observers? Explain. b) Imagine you are the pedestrian in the street, how would you observe the other four observers? Explain. c) Based on your answers above, explain what it means when someone says an object is moving. d) Consider the phrase motion is relative. Use your idea of what it means to move to explain the meaning of this statement. 1.3 Represent and Reason Examine the map below. Your friend is visiting Washington DC and is staying at George Washington University. Your friend must walk across town to get to the Smithsonian Institution. PUM Kinematics Lesson 1: Essentials to Describing Motion 5

6 a) Provide simple directions to your friend so she can get to the Smithsonian by 8:30 am. b) What was the object of reference that you selected to provide directions? c) How did the map legend help you to give essential directions? d) What were the assumptions you made when you provided directions? Need Some Help? Assumptions are issues we take for granted in an experiment. They can help us explain why the results weren t exactly as we expected or could be something that wasn t taken into account when we designed the experiment. Despite the outcome of the experiment, it s important to consider the assumptions that may have been made. Assumptions are factors that could affect our results, but have not been included in our calculations or reasoning. e) How did this affect the time you tell your friend to leave? Did You Know? Motion: An object is in motion with respect to another object (reference object) if, as time progresses, its position is changing relative to the reference object. Reference frame: A reference frame includes three essential components: 6 PUM Kinematics Lesson 1: Essentials to Describing Motion

7 1. An object of reference, which is a real object in the physical world. 2. A clearly defined coordinate system. The coordinate system includes labels for the direction of the axis, such as north, south, east, west, left, right, up down, or positive and negative. The unit scale for measuring distances is also identified. A point on the coordinate system, usually the 0 point, is attached to the object of reference. 3. A zero clock reading that serves as a reference for future clock readings. 1.4 Reason Meagan and Beccy are sitting in a train, which is carrying them east toward New York City. They see Ryan walking down the aisle toward the rear of the train. He is texting on his phone and does not see that he's about to bump into Andrew, who is standing in the aisle. Meagan says "Ryan must moving west, because he is about to bump into Andrew." Beccy says "That can't be true, because if Ryan was moving west he would never get to New York City." Meagan says "But he can't be moving east or else he would not be about to bump into Andrew!" a) Act out the situation. b) Do you think Meagan or Beccy is correct? Is either one wrong? Explain how you evaluate their reasoning. c) Explain carefully what is REALLY correct about the motion of Ryan. 1.5 Relate Homework Describe three situations from your life that are relevant to your idea of relative motion and reference frames. 1.6 Explain a) Using the idea of a reference frame provide directions to get to school from your house 15 minutes before homeroom begins. b) When we say the Sun is rising and the Sun is setting, who is the observer? Where should the observer be to see that the Sun does not move around the Earth every 24 hours? c) Someone in your class says the Sun does not move, then why do we have days and nights? PUM Kinematics Lesson 1: Essentials to Describing Motion 7

8 1.7 Observe and Explain a) Use what you have learned so far about describing motion to explain the sequence of photographs using two different reference frames. Photo 1 Photo 2 b) Imagine that you were an observer somewhere in the picture above, where should you be located to see: the student moving? the student not moving? the street sign moving? the police car moving right? the police car moving left? the bicycle moving? the bicycle not moving? c) From photos 2 and 3 above, where would an observer have to be located so that she sees the pickup truck moving backwards? Explain why this is the case. Reflect: What did you learn during this lesson? Can you explain to your friends or relatives what the words motion is relative mean? What is a reference frame and why is it important to know about it? 8 PUM Kinematics Lesson 1: Essentials to Describing Motion

9 2.1 Observe and represent Lesson 2: Describing Motion Scientists communicate with each other using pictures, words, mathematical relations, graphs, and many other representations. In this observational experiment, you are going to learn how to represent motion in various ways. Before you begin, place a ball on the metal track. Practice rolling the ball along the track with one of your teammates. a) Describe the motion of the ball as best you can. b) With your teammates, think of a way to keep the clock reading and mark the location of the ball for each second as it rolls. You can use any marking method you think is appropriate. Describe your method and practice until you are comfortable. Here s an Idea! - To practice your counting look at a second hand on a stopwatch while tapping on a desk or shouting a word. If you yell out, make sure that when you shout your counts for each second, they are brief; for example yes. c) Once your team decides on a procedure, perform the experiment. d) Designate an origin and record each clock reading and position for each mark: be sure to include the axis and designate directions. e) Revise your description in part (a). f) Explain how you can use the marks to describe the motion of the ball 2.2 Hypothesize Think about how the motion of the ball and the marks are related in general. Propose your hypothesis here. 2.3 Test Your Idea Imagine that you now let the ball roll down the track that is tilted with respect to he horizontal. Use the hypothesis in 2.2 to predict the relative spacing of the marks for the ball rolling down the incline. PUM Kinematics Lesson 2: Describing Motion 9

10 Scientific Ability Rubric to self-assess your prediction Missing An attempt Needs some improvement Acceptable Is able to distinguish between a hypothesis and a prediction. No prediction is made. The experiment is not treated as a testing experiment. A prediction is made but it is identical to the hypothesis. A prediction is made and is distinct from the hypothesis but does not describe the outcome of the designed experiment. A prediction is made, is distinct from the hypothesis, and describes the outcome of the designed experiment. Perform the experiment and compare your prediction to the outcome of the testing experiment? What does this tell you about your hypothesis? Did You Know? One representation of motion is called a dot diagram. To make a dot diagram mark locations of a moving object at equal time intervals. 2.4 Reason a) Imagine that you could observe an airplane flying from New York to Los Angeles. The length of the airplane is 32 m. The length of the trip is about 4000 km or m. The average speed of the plane is 500 km/h. You need to calculate the time that it takes the airplane to fly from NY to LA. Do you need to take the size of the plane into account? Explain. b) While traveling to the gate the plane turns left. In order to describe how much a passenger moved with respect to the ground, do you need to take the size of the airplane into account? Explain. c) A passenger is sitting on the plane. The moves plane moves forward 100 m. Do you need to take the size of the airplane into account in order to determine how far the passenger moved? Explain. d) The same plane parks at the gate. You need to calculate how close the gates can be to each other for the airplanes to park safely. Do you need to take the size of the plane into account? Explain. e) When is it important to take the size (or dimensions) of the plane into account? Did you know? Point-like object: When you do not need to take the size of the object into account to solve a problem, you can represent this object as a point. This point will have all properties of the object except its size and parts. This simplified object is called a 10 PUM Kinematics Lesson 2: Describing Motion

11 point-like object. A point-like object is a simplified version of a real object. We consider real objects to be tiny point-like objects under two circumstances: (a) when all their parts move in same way, or (b) when the objects are much smaller than the dimensions of the process described in the problem. [or] when the size and shape of the object is not particularly important to the situation. The same object can be modeled as a point like object in some situations and not in others. f) How did a point like object play a role in the experiments at the beginning of lesson 2? Homework 2.5 Represent and reason Below there are images of snail crawling along a table next to a ruler. Clock reading = 0 s Clock reading = 1 s Clock reading = 2 s Clock reading = 3 s Clock reading = 4 s Photographs like the ones seen above can be used to study the motion of objects. a) Determine the reference frame for these photographs. b) Record the position of the snail for each clock reading. What assumption(s) did you make? PUM Kinematics Lesson 2: Describing Motion 11

12 c) Create a dot diagram for the motion of the snail. d) How would your picture look different if the motion happened over 8 seconds rather than 4 sec? 2.6 Represent and Reason a) Meg created a dot diagram for a bug she found on the ground. Use the dot diagram to describe the motion of the bug. Is it moving at a constant pace, is it speeding up or slowing down? Explain. What would to be true about an observer that draws the same dot diagram as Meg? have b) The dot diagrams for the two bicyclists are show below. When were they at the same location at the same time (circle)? 2.7 Represent and Reason Bike 1 Marks on the road I I I I I I I I 20m 60m 100 m 140m Bike 2 Imagine you are taking a road trip to Cape May, NJ (the Southern most part of NJ) and you program the directions into your GPS. The GPS says that it will take you 2 hours to travel. When you leave, it is 8:00 am but by the time you arrive it is 10:45 am. What assumptions did the GPS program make about the trip to Cape May? What assumptions may have been incorrect? Explain. If you did arrive at Cape May at 10:05 am, were the assumptions the GPS system made fair? Explain. Reflect: What did you learn during this lesson? Can you explain to your friends or relatives why a dot diagram is useful and why we care about pointlike models of objects? 12 PUM Kinematics Lesson 2: Describing Motion

13 Lesson 3: Reasoning in Kinematics: Inventing an Index An index is a number that helps people compare things. Miles per gallon is an index of how well a car uses gas. Batting average is an index of how well a baseball player hits. Grades are an index of how well students perform on a test. We want you to invent a procedure for computing an index that helps make comparisons. RK.1 The Popping Index Three companies make popcorn. They use different types of corn so the popping is fast or slow. Invent a procedure for computing a popping index to let consumers know how fast each brand pops. Rules for the Index 1. The same brand of popcorn pops at the same speed. So a brand of popcorn only gets a single popping index. 2. You have to use the exact same procedure for each brand to find its index. 3. A big index value should mean that the popcorn pops faster. A small index number should mean that the popcorn pops slower. PUM Kinematics Lesson 3: Reasoning in Kinematics: Inventing an Index 13

14 14 PUM Kinematics Lesson 3: Reasoning in Kinematics: Inventing an Index

15 RK2. Fastness Index Let s look at another kind of index. Your task this time is to come up with a fastness index for cars with dripping oil. You will see a bunch of cars, and you need to come up with one number to stand for each car s fastness. There is no watch or clock to tell you how long each car has been going. However, all the cars drip oil once a second. (They are not very good cars!) Oil You can look at the oil drops to help figure out how long a car has been traveling. This task is a little harder than before. A company always makes its cars go the same fastness. We will not tell you how many companies there are. You have to decide which cars are from the same company. They may look different! To review: (1) Make a fastness index for each car. (2) Decide how many companies there are. (3) To show the cars that are from the same company, draw a line that connects the cars. Start A Start B Start C Start D Start E Start F PUM Kinematics Lesson 3: Reasoning in Kinematics: Inventing an Index 15

16 RK.3 Reason 1. Which popcorn is fastest? Give an explanation why you picked that popcorn. 2. Which cars are fastest? Give an explanation why you picked that car. 3. For each question below, describe the steps that you would take to get an answer: A full bowl of popcorn has 60 popped corns in it. How long does it take the fastest popcorn to fill a bowl of popcorn? How long does it take for the fastest car to travel 15 blocks? How many popped corns will the fastest popcorn pop in 70 seconds? How far will the fastest car travel in 20 seconds? 4. Another company, Acme, has an index of 2.5. a) Let s say that Acme makes popcorn. Using everyday language, describe the specific information that the number 2.5 tells about this their popcorn. b) Make a sketch that explains your answer to part a). c) Now let s say that Acme makes cars. Using everyday language, describe the specific information that the number 2.5 tells about their car. d) Make a sketch that explains your answer to part c). 5. Explain why you think there were less than 6 car companies, even though there were six different diagrams describing the car companies. RK.4 The Steepness Index Let s try another kind of index. As the owner of 2-Die-4 Water Park, you are in charge of buying the slides. Most of your clients are teenagers, and they like the steepest slides they can find. You want to buy slides that will attract the most business. You are trying to choose between the slides shown below. Invent a procedure for computing a steepness index so that you can buy the best slides, and prove to your customers that you have the steepest slides in town. Rules for the Index 1. Each slide gets one steepness index because it has the same steepness all the way down. 2. You have to use the exact same procedure for each slide to find its index. 3. A big index value should mean that the slide is steep. A small index number should mean that the slide is less steep. 16 PUM Kinematics Lesson 3: Reasoning in Kinematics: Inventing an Index

17 Find the steepness index for the following portions of these slides 16 ft 18 ft 20 ft 12 ft 18 ft 7 ft 5 ft 24 ft Rocks Ur Socks Splash Attack Super Soaker Tsunami RK5.Reason 1. Which slide is steepest? Give an explanation why you picked that slide. 2. For each question below, describe the steps that you would take to get an answer: In another portion of the steepest slide, it drops down by 14 feet. How far does the slider move horizontally when she goes down that part of the slide? How far down has she dropped when she has moved horizontally 5 feet? 3. Another company, Acme, has an index of a) Let s say that Acme makes slides and 0.75 is the steepness index. Using everyday language, describe the specific information that the number 0.75 tells about their slides. b) Make a sketch that explains your answer to part a). 4. A quality control officer for Hop-On Popcorn counted the number of kernels popped at several different times. The data are shown below in the table. Popped Kernels Time (sec) a) Make a graph of the popped kernels vs. time. b) Find the steepness index of the best-fit line. What information does the steepness of the best-fit line tell you about Hop-On Popcorn? c) Which pops faster, the Hop-on Popcorn or HipHop Popping Corn? Explain. d) Sketch a bowl of Hop-On Popcorn after 15 seconds in an air popper. PUM Kinematics Lesson 3: Reasoning in Kinematics: Inventing an Index 17

18 3.1 Observe and represent Lesson 3A: How Fast Are You Moving? Roll out a 10 m tape measure across the classroom floor. The teacher will assign each group a different starting position and direction for their cars (a fast and slow car). For each car, you will mark the position along the tape measure at each second. Make sure the cars travel at two different rates. a) Discuss with your teammates and decide on a method for keeping time and marking the location of the car each second. When you are ready, start with Clock Reading t Slow Car Position x your slow car; record the data in the table below. Repeat the experiment for the fast car and record the data in the table below. Be sure the position represents the location of the car with respect to the tape measure on the floor. b) What are the physical quantities measured in this experiment? What units of measure did you use? c) Explain the differences between these two ideas. Clock Reading t Fast Car Position x Did You Know? Physical Quantity: A physical quantity is a characteristic of a physical phenomenon that can be measured. A measuring instrument is used to make a quantitative comparison of this characteristic and a unit of measure. Examples of physical quantities are your height, the speed of your car, or the temperature of air or water. If a characteristic does not have a unit, it is not a physical quantity. Position x is the location of an object relative to a chosen zero on the coordinate axis. Time Interval: The time interval is the difference between two clock readings. If we represent one time reading as t 1 and another reading as t 2, then the time interval between those two clock readings is t 2 - t 1. Another way of writing this statement is: t 2 t 1 = t The symbol is the Greek letter delta and in physics and mathematics it is read as delta t ( t) or the change in t. Time can be measured in many different units, such as seconds, minutes, hours, days, years, and centuries, etc. 18 PUM Kinematics Lesson 3A: How Fast Are You Moving?

19 Clock reading: The clock reading or time (t) is the reading on a clock, on a stopwatch, or on some other time measuring instrument. Example: You start observing the motion of your car when stop watch shows 3 seconds, when you finish the watch shows 15 seconds. What is the time interval for your observations? For your observations t 1 =3 seconds t 2 = 15 seconds, thus t 2 t 1 = t = 15 3 = 12 s. Here the physical quantity is t and the units are seconds. 3.2 Represent and Reason a) Robin, James, Tara and Joe (at rest with respect to each other) collected data for the motion of the same car. They each represented the data differently. Examine the four representations below; select a representation that would best represent the position of the car as a function of time. Explain. Robin Tara PUM Kinematics Lesson 3A: How Fast Are You Moving? 19

20 James Joe b) Discuss your choice and reasoning with the class. c) Represent the motion of the cars with a graph (plot the data from each car on the same axes) using the data collected in activity. The position of the car is recorded on the vertical axis and the clock reading on the horizontal axis. Did You Know? In science, experimenters put time on the horizontal axis when it is the independent variable. d) Draw a trend line for each car on the graph you drew in part (c). What information can you learn about the motion of the car from the graph? Explain. Need Some Help? Trend Line: A trend line represents a trend in the data. To draw a trend line, try to draw a smooth line that passes as close to all data points as possible. The data points do not need to be on the line. 3.3 Represent and Reason a) Compare the trend lines of the two cars? How are they different? b) Find the slopes of the two lines. Explain how you found the slope. What name could you give to the slope? c) Explain what it means if the slope is positive or negative. 20 PUM Kinematics Lesson 3A: How Fast Are You Moving?

21 Did You Know? Velocity of an object moving at constant velocity is the slope of the position versus time graph and is equal to the change in position of the object divided by the time interval during which this change in position occurred. When the object is moving at constant velocity this ratio is the same for any time interval v = x 2 x 1 t 2 t 1 = x t where x 2 x 1 ( x) is any change in position during the corresponding time interval t 2 t 1 ( t ). The unit for velocity is m/s, miles/h, km/h, and so forth. Positive velocity means that the object is moving in the positive direction; negative velocity means it is moving in the negative direction. Speed is the magnitude of velocity, it is always positive. d) What is the velocity of each car in your experiment? How did you do that? e) How is velocity related to the concept of index you invented in lesson 3? f) Write a function x(t) for the fast car and a separate function for the slow car. What role does the trend line play and what role does the y-intercept play in writing this function? Need Some Help? When mathematicians and physicists express patterns mathematically they use functions. A function is a rule that one uses to find a dependent variable when an independent variable is known. You may have met functions in a math class. There the independent variable was labeled x and the dependent variable is labeled y. The function then is y(x). In science and math class you can actually use any labels as long as you agree on which was the independent and which is the dependent variable. For the problem below, the independent is t, and a dependent is x. Example: Examine: Define: Represent: Describe the relationship between the two variables. Describe the variables used in the scenario Write a mathematical equation using variables The object changes its position by 50 meters each second Time (second) Position (meters) t = time elapsed x = position x = 50t This expression can all be written in function notion as x(t) = 50t, however, in physics it is necessary to include units of measure x(t) = 50(m/s) t or x(t)=50(m/s)t. x(t) is read as x of t. PUM Kinematics Lesson 3A: How Fast Are You Moving? 21

22 REMEMBER! When you graph a function the independent variable is always placed on the horizontal and the dependent variable on the vertical axis. Position is on the vertical axis because position "depends on" clock reading. 3.4 Practice Homework a) A car moved from x 1 = 20 mi to x 2 = 62 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 and x 2, and find x. b) A car moved from x 1 = 120 mi to x 2 = 34 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 and x 2, and find x. c) A car moved from x 1 = 30 mi to x 2 = 62 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 and x 2, and find x. d) A car moved from x 1 = 20 mi to x 2 = 78 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 and x 2, and find x. e) A car moved from x 1 = 62 mi to x 2 = 20 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 and x 2, and find x. 3.5 Practice In the previous example, the time interval during which the position change occurred t = 1.5hr. Determine the velocity and the speed of the car for each x. What does it mean if velocity is positive? Negative? To answer relate to the direction of the x axis. 3.6 Analyze a) The graph below shows the motion of a football player during 20 seconds. What is the player s position at the point shown with the triangle on the graph? Choose the answer that you think is best. I) 2.5 yards; II) 10 yards; III) 35 yards; IV) 25 yards. b) How far did the player travel from the beginning of observations? I) 2.5 yards; II) 20 yards; III) 35 yards; IV) 25 yards. c) What happened at the 0 clock reading: 22 PUM Kinematics Lesson 3A: How Fast Are You Moving?

23 I) The player started moving; II) The player was passing the mark of 45 yards; III) The player was moving in the negative direction; IV) both II and III are correct. d) Which answer best describes the player s motion at the point indicated by the triangle on the graph? I) The player is moving at constant speed: II) The player encountered a dip and is moving slightly downhill; III) The player is slowing down; IV) The player stopped. 3.7 Represent and Reason Two objects are moving in the same direction. The speed of one is 5 m/s and the speed of the other is 10 m/s. When you start observing them, they pass the same location at the same time. a) Draw dot diagrams for two objects. b) Represent their motions with position versus time graphs. Use the same scale for both objects. b) Choose from position versus time functions describing their motions a combination that looks correct to you: I) x 1 = (5 m/s) + t; x 2 = (10 m/s) + t; II) x 1 = (5 m/s)t; x 2 = (10 m/s)t; III) x 1 = t(5 m/s); x 2 = t(10 m/s); IV) both II and III are correct. c) How long will it take each object to travel 276 m? PUM Kinematics Lesson 3A: How Fast Are You Moving? 23

24 d) How far from each other will they be in 10 seconds? 20 seconds after you start observing them? 3.8 Represent and Reason The motion of object A is represented by the function x A = (5.0 m/s)t ; the motion of object be is represented by the function x B = ( 3.5 m/s)t. a) Say everything you can about the motions of those objects. If you need to assume something, state your assumption clearly. b) Represent the motions in as many different ways as you can. 3.9 Practice A train is moving at the speed of 15 m/s. How far will it move in 10 seconds? In 10 minutes? In 10 hours? 3.10 Practice You are riding a bicycle to your friend s house. The house is 3 km away. You arrive at the house in 17 minutes. a) What was your speed? Write the speed in km/min; in m/s; and in mph. List all assumptions that you made. b) Write a function x(t) for your ride. In how many ways can you write this function? 3.11 Practice Usually, a briskly-walking person can cover 4 miles in an hour. How long will it take this person to walk 12 miles? 0.3 miles? 4 kilometers? What assumptions did you make? 3.12 Reason You walk 1.8 miles every 30 min. Use the index approach to calculate in your head how far you will walk in: (1) 1 hour; (b) 1 hour 30 minutes; (c) 2 hours. Reflect: What did you learn in this lesson? Can you explain to your friends how the slope of the position versus time function is related to the object s velocity? What does it mean if velocity is positive? What does it mean if it negative? 24 PUM Kinematics Lesson 3A: How Fast Are You Moving?

25 4.1 Hypothesize Lesson 4: The Moving Man An object is moving in the positive direction at constant velocity v. It starts at clock reading t = 0 sec, at a position x 0. How would you write a function that will allow you to find the position of the object at any time? 4.2 Test Your Idea with Phet Simulations Go to and click on the simulation The Moving Man. Browse to (or google for phet moving man) and click Run Now You should see a screen like the one shown. There is a man at top of the simulation who can move 10 m in either direction from the origin. The simulation also includes axes of position, velocity and acceleration graphs that will reflect his motion. Since you are not going to use the acceleration or velocity graph right away, you can close them by clicking on the small window in the upper right hand corner of each section. To eliminate the walls, click on special features, then click on free range. Use the hypothesized mathematical model in activity 4.1 to predict the position. Scenario 1: The man s initial position is 9 m and he is jogging to the left at 2 m/s. a) Write an expression for the man s position as a function of time. b) Create a position vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the time when he passes through the position at 0m. e) Perform the experiment by entering given quantities in the respective simulation boxes and click Go! Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. Scenario 2: The man is walking at the speed of 0.75 m/s towards his home. When we start observing him, he is at the position of 7 m to the left of the origin. a) Write an expression for the man s position as a function of time. PUM Kinematics Lesson 4: The Moving Man 25

26 b) Create a position vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the time when he arrives at the house. e) Perform the experiment through the simulation. Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. Scenario 3: When we start observing the man is at the 5 m mark by the house and is running at the speed of 4.5 m/s towards the tree. a) Write an expression for the man s position as a function of time. b) Create a position vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the time when he has traveled 70 m beyond the tree. e) Perform the experiment through the simulation. Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. 4.3 Test Your Idea Use your newly modified hypotheses from the previous activity to predict how you d have to move so that a motion detector creates position versus time graphs that match the graphs in the previous activities. Explain how your prediction compares to the outcome. 4.4 Represent and Reason In this activity you will be acting as the Moving Man or Woman. Below are a number of position vs. time functions and written descriptions. Here you will have to act out how to move so that a motion detector creates position versus time graphs match description provided. Before acting it out, discuss how your motion should match the written and mathematical descriptions. a) A person is 7.0 m away when we start observing and walks towards the origin at 0.4 m/s. 26 PUM Kinematics Lesson 4: The Moving Man

27 b) You are 5.0 m away from the origin at the initial clock reading and walk at 1m/s for 4 seconds towards the origin stop for 3 seconds then walk 0.5 m/s for 6 seconds away from the origin. c) x(t) = 5m +( 0.7m/s)t d) x(t) = 0.8m + (1.2m/s)t 4.5 Represent and reason Homework m -5.0 m 0.0 m +5.0 m m Examine the dot diagram above. When we start observing the object it is at +7.5 m and moves in the negative direction of the x-axis. a) Describe the motion in words. b) Sketch a position vs. clock reading graph. c) Write a function for the position as a function of time for the object s motion. 4.6 Represent and Reason Thus far, you have represented the motion of a ball with different representations: dot diagrams, words, pictures, tables, and now a graph. Explain how the different representations describe the same motion. a) Use the graph below to describe the motion in words. Pay attention to what happened at zero clock reading! b) Write a function for the position as a function of time for the object s motion. c) You should notice that the two physical quantities on the graph do not have units. Describe a real life situation for this motion if the units were kilometers and seconds. Describe another situation if the units were centimeters and minutes. d) Draw a picture for each of the situations you described in part c. PUM Kinematics Lesson 4: The Moving Man 27

28 4.7 Equation Jeopardy Three situations involving constant velocity are described mathematically below. 1. ( 86 m) = v(1.72 s) + ( 100 m) 2. x = ( 5.7 m/s)(300 s) + (1000 m) 3. ( m) = ( 5.7 m/s)(6.8 s) + x initial a) Write a story or a word problem for which the equation is a solution. There is more than one possible problem for each situation. b) Sketch a situation that the mathematical representation might describe. c) Determine the unknown physical quantity. 4.8 Practice You are learning to drive. To pass the test you need to be able to convert between different speedometer readings. The speedometer says 65 mph. (a) Use as many different units as possible to represent the speed of the car. (b) If the speedometer says 100 km/h, what is the car s speed in mph? 4.9 Practice The speed limit on the roads in Russia is 75 km/h. How does this compare to the speed limit on some US roads of 55 mph? 4.10 Practice Convert the following record speeds so that they are in mph, km/h, and m/s. (a) Australian dragonfly 36 mph; (b) the diving Peregrine falcon 349 km/h; and (c) the Lockheed SR-71 jet aircraft 980 m/s (about three times the speed of sound) Reason You are moving on a bicycle trying to maintain a constant pace. You cover 23 miles in 2 hours. What is your speed in m/s? If you only rode half of the distance maintaining the same pace what would the speed be? If you rode 43 miles, what would the speed be? 4.12 Reason James and Tara argue about speed. James says that the speed is proportional to the distance and inversely proportional to the time during which the distance was covered. Tara says that the speed does not depend on the time or distance. Why would each say what they did? Do you agree with James? Do you agree with Tara? How can you modify their statements so that you could agree with both of them? 4.13 Hair growth speed Physicists often do what is called order of magnitude estimations. Such estimations are approximate calculations of some quantity that they are interested in. For example, how do we estimate the rate that your hair grows in mm/s? 28 PUM Kinematics Lesson 4: The Moving Man

29 Think of the following: How often do you get haircuts? How long does your hair grows during this time? Then convert the time between hair cuts to seconds and the length of your hair growth to millimeters then you are almost done. The question is how will you report your results? What if after dividing length by time you get a number on your calculator that looks like ? Think how you can report the result so it looks reasonable. Did You Know? Significant digits When we measure a physical quantity, the instrument we use and the circumstances under which we measure it determine how precisely we know the value of that quantity. Imagine that you wear a pedometer (a device that measures the number of steps that you take) and wish to determine the number of steps on average that you make per minute. You walk for 26 min (as indicated by your wristwatch) and see that the pedometer shows 2254 steps. You divide 2254 by 26 using your calculator and it says If you accept this number, it means that you know the number of steps per minute within plus or minus steps/minute. If you accept the number 86.69, it means that you know the number of steps to within 0.01 steps/minute. If you accept the number 90, it means that you know the number of steps within 10 steps/minute. Which answer should you use? The number of the significant digits in the final answer should be the same as the number of significant digits of the quantity used in the calculation that has the smallest number significant digits. Thus, in our example, the average number of steps per minute should be 86, plus or minus 1 steps/minute: 86±1. In summary the precision of the value of a physical quantity is determined by one of two cases. If the quantity is measured by an instrument, then its precision depends on the instrument used to measure it. If the quantity is calculated from other measured quantities, then its precision depends on the least precise instrument out of all instruments used to measure a quantity used in the calculation. Another issue with significant digits arises when a quantity is reported with no decimal points. For example, how many significant digits does 6500 have two or four? This is where the scientific notation helps. Scientific notation means writing numbers in terms of their power of 10. Example: we can write 6500 as 6.5 x This means that the 6500 actually has two significant digits. If we write 6500 as 6.50 x 10 3 it means 6500 had PUM Kinematics Lesson 4: The Moving Man 29

30 three significant digits. Scientific notation provides a compact way of writing large and small numbers and also allows us to indicate unambiguously the number of significant digits a quantity has. [This is great too, could add some practice later] 4.14 Evaluate On the web you might find the flowing statement: The speed of hair growth is roughly 1.25 centimeters or 0.5 inches per month, being about 15 centimeters or 6 inches per year. With age the speed of hair growth might slow down to as little as 0.25 cm or 0.1 inch a month. Is this result consistent with your estimate? Are the significant figures reported in for different measurements consistent with each other? 30 PUM Kinematics Lesson 4: The Moving Man

31 Lesson 5: How Many Velocities can an Object Have? 5.1 Observe and Reason Examine the Position vs. Clock Reading graph for the football player below. a) What were the player s positions at the points shown with triangles on the graph? b) Describe the motion of the player in words. Act it out. Pay attention to what happened at 0 clock reading! c) James said the football player traveled 5 yards in the negative direction. Tara said the football player moved 85 yards total. Joe said the football player traveled 5 yards. How did each person arrive at his/her answer? d) What is the distance traveled by the football player from t = 1.0 s to t = 12.0 s? What is the path length traveled by the football player travel from t = 1.0 s to t = 12.0 s? What is the displacement of the football player travel from t = 1.0 s to t = 12.0 s? e) Explain why the values for the quantities in d) are written as t = 1.0 s for example as opposed to t = 1 s? PUM Kinematics Lesson 5: How Many Velocities can an Object Have? 31

32 Did You Know? Position, displacement, distance and path length: Position x is the location of an object relative to a chosen zero on the coordinate axis. Displacement x 2 - x 1 indicates a change in position from clock reading t 2 to clock reading t 1. The sign of the displacement indicates the direction of the displacement (+ when the object moves in the positive direction as x 2 > x 1 and when the object moves in the negative direction of the chosen axis as x 2 < x 1 ). The magnitude of that position change is called the distance. It is always positive. The path length is the length of the path that the object traveled. If the object returns to the same point where it started, the displacement and distance are zero, but the path length is not. f) What is the distance traveled by the football player travel from t = 2.0 s to t = 19.0 s? What is the path length traveled by the football player from t 6 = 2.0 s to t 7 = 19.0 s? What is the displacement of the football player from t = 2.0 s to t = 19.0 s? Explain a) Devise a method for obtaining the value for displacement, distance and path length on a position versus clock reading graph. b) Your friend says that to find displacement, he needs to take the position reading at point 2 and subtract the position reading at point 1. Do you agree or disagree? 5.3 Observe and Reason Examine the Position vs. Clock Reading graph for the same football player running the same play below. a) What were the player s positions at the points shown with triangles on the graph? 32 PUM Kinematics Lesson 5: How Many Velocities can an Object Have?

33 b) Examine the graph above. What do you know about this observer compared to the original observer? 5.4 Represent and Reason A car stops for a red light. The light turns green and the car moves forward for 3 seconds at a steadily increasing speed. During this time, it travels 20 meters. The car then travels at a constant speed for another 3 seconds for a distance of 30 meters. Finally, when approaching another red light, the car steadily slows to a stop during the next 3 s in 15 meters. a) What is the total path length that the car traveled? b) What is the average speed of the car? Need Some Help? To find the average speed, you need to divide the total path length traveled by the total time of travel. c) How does the total average speed for the entire 9 seconds compare to the average speed for each of the 3-second intervals? Why are the average speeds different? Explain. d) What is the average velocity? e) A car traveled for 15 s at 10 m/s and another 15 s at 20 m/s. What is the average speed? f) The same car traveled 200 m at 10 m/s and another 200 m at 20 m/s. What is the average speed? Discuss the difference between the results here and in part (e). 5.5 Represent and Reason Homework A bus filled with physics students going to Great Adventure for Physics day travels 280 km West along a straight-line path at an average velocity of 88 km/hr to the west. The bus stops for 24 min, then it travels 210 km south with an average velocity of 75 km/hr to the south. a) Diagram and label all the pertinent information for this trip. b) What is the average velocity for the total trip? What is the average speed for the total trip? PUM Kinematics Lesson 5: How Many Velocities can an Object Have? 33

34 5.6 Represent and Reason The position of an object is represented in the graph above. a) Describe the motion in words. b) What is the average velocity of the object for the different time intervals: 0-10 sec and sec? c) What is the average speed for the object during the entire 30 sec? What is the average velocity during that same interval? d) What is the average speed and average velocity for the time interval from 5 sec 25 sec? 5.7 Represent and Reason 200 m mark 60 m 100m Start / Finish The picture above is a diagram of a 400m outdoor track. All races begin at the start/finish line. 34 PUM Kinematics Lesson 5: How Many Velocities can an Object Have?

35 a) If the 1600m race is 4 laps, what is the path length raced? What is the displacement? b) The 200 m run begins at the 200 m mark and finishes at the start/finish line, what is the path length raced? What is the magnitude (the amount) of the displacement, or distance raced? 5.8 Represent and Reason In the table on the right you have data about the moving object. How far did the object travel in the 10 seconds? I) 230 m; II) 70 m; III) 180 m. Time (second) Position (meters) Reflect: What did you learn in this lesson? Why is it important to know about average velocity and average speed? How is possible for a car to have an average velocity of zero and average speed of 65 mph? Give an example. PUM Kinematics Lesson 5: How Many Velocities can an Object Have? 35

36 Lesson 6: Representing the same thing in different ways 6.1 Observe and Represent Two cars are let go simultaneously on a smooth floor. You and a friend follow each car and drop a beanbag every second to mark the location of your car at every time interval. Car 1 Car 2 Describe the motion of each car as fully as possible by answering the following questions: a) Were the cars ever next to each other? If so, how do you know? At what clock reading(s) does it happen? Explain. b) Eugenia answered the question above by drawing the following figure: Car 1 Car 2 Why do you think she circled the dots? How can you help her understand your point of view? c) Which direction are the cars moving? d) Using arrows, how would you make the picture above a more complete representation of motion? e) If a general x(t) equation for the first car is x 1 (t) = v 1 t. What would be the same about the equation for the second car? What would be different? 6.2 Represent and Reason Madeline and Gabi walked eastward on a marked path at a velocity of 0.7 m/s. The path runs East and West with every 1000 m marked. If they begin at the mark of 6000 m and walked for 25 min, how far have they walked? At what mark would they be at the end of the walk? 6.3 Represent and Reason Darshan, roller skating down a marked sidewalk, was observed to be at the following positions at the times listed below. Answer the following questions a) Plot a position vs. time graph for Darshan s motion. b) Determine the time it will take for Darshan to skate to position -85.0m. Be sure to discuss assumptions made. t (s) x (m) PUM Kinematics Lesson 6: Representing the same thing in different ways

37 c) Draw a dot diagram for the 10-second time interval. d) How far did Darshan travel in 5 seconds: I) 10.0 m; II) m; III) m. 6.4 Equation Jeopardy A situation involving constant velocity is described mathematically below. 114mi = ( 62 mi hr )(0.35hr) + x 0 a) Sketch a situation that the mathematical representation might describe. There is more than one possible situation for the equation. Pay special attention to what happens at the zero clock reading. b) Write in words a problem for which the equation is a solution. c) Draw a dot diagram for the motion. d) Plot a position vs. time graph for this motion. e) Determine the unknown quantity. 6.5 Represent and Reason Calculate the average speed and average velocity of a complete round trip in which a train travels 200 km at 90 km/hr, stops for an 1.0 hour and returns back to the starting point at 50 km/hr. a) Plot a position vs. time graph for this motion. Identify the place on the graph that represents the hour break. b) Solve for as many unknown quantities as possible. 6.6 Practice a) A car moved for 30 min at the speed of 55 mph and for another 30 min at 75 mph. What was the average speed of the car? b) A car moved for 30 miles at the speed of 55 mph and for another 30 miles at 75 mph. What was the average speed of the car? PUM Kinematics Lesson 6: Representing the same thing in different ways 37

38 6.7 Equation Jeopardy Homework A situation involving constant velocity is described mathematically below. x = ( 62.0 mi )(0.35hr) mi hr a) Sketch a situation that the mathematical representation might describe. There is more than one possible situation for the equation. b) Write in words a problem for which the equation is a solution. c) Draw a dot diagram for the motion. d) Plot a position vs. time graph for this motion. e) Determine the unknown quantity. f) Repeat the same steps for the equation: 114mi = ( v mi )(1.2hr) + ( 30.0 mi). hr 6.8 Practice (very challenging!) a) A car moved half of the time at the speed of 55 mph and for the other half at 75 mph. What was the average speed of the car? b) A car moved for half the distance at the speed of 55 mph and the other half distance at 75 mph. What was the average speed of the car? Here s an Idea! Physics is about problem solving. Every problem is different, however some general strategies might help. Examine the steps below they might not be relevant for every problem you encounter, but some of them are useful all the time. Problem Solving Strategy Sketch and Translate: Read the text of the problem at least three times to make sure you understand what the problem is saying. Visualize the problem, make sure you see what is happening. Sketch the situation described in the problem. Include an object of reference, a coordinate system and indicate the origin and the positive direction. Label the sketch with relevant information. 38 PUM Kinematics Lesson 6: Representing the same thing in different ways

39 Draw a physical representation: Think of whether a dot diagram or a graph will help you understand the problem. Represent the problem situation with either or both. Represent Mathematically: Use the sketch(s), diagram(s), and graph(s) to help construct a mathematical representation (equations) of the process. Be sure to consider the sign of each quantity. Solve and Evaluate: Solve the equations to find the answer to the question you are investigatin Evaluate the results to see if they are reasonable. To do this, check the units, decide if the calculated quantities have reasonable values (sign, magnitude), and check limiting cases. Go bck to the sketch and the physical representation to make sure your answer is consistent with both. Do not rush! Reflect: What did you learn during this lesson? If you were to write a letter to your past self about different representations, what would you say? Why are different representations important in physics? Which representation are you most comfortable with? Which representation gives you the most trouble? After you found the latter, make sure you focus on them in the future lessons. PUM Kinematics Lesson 6: Representing the same thing in different ways 39

40 Lesson 7: Using Velocity-versus-Time Graphs to find Displacement 7.1 Represent and Reason The figure on the right shows a velocity-versus-clock reading graph that represents the motion of a bicycle, modeled as a point particle (we are not interested in the motion of the feet, head, etc.), moving along a straight bike path. The positive direction of the position axis is toward the east. a) Describe the motion of the bike in words. b) How is this graph different from the position versus time graph for the same motion? c) Think of how you can use the graph to estimate the bike s displacement from a clock reading of 10 s to a clock reading of 15 s. Explain. (Hint: Think of the area of a rectangle.) d) Use the graph to estimate the bike s displacement from a clock reading of 0 s to 20 s. e) Formulate a general rule for using a velocity versus clock reading graph to determine an object s displacement during some time interval if the object is moving at constant velocity. f) Is this rule consistent with the mathematical models developed in the previous lessons x(t) = x 0 + vt for constant velocity? 7.2 Represent and Reason The figure below shows a position-versus-clock reading graph that represents the motion of a bicycle, modeled as a point like object (we are not interested in the motion of the feet, head, etc.), moving along a straight bike path. The positive direction of the position axis is toward the east. a) Describe the motion of the bike in words. 40 PUM Kinematics Lesson 7: Using Velocity-versus-Time Graphs to find Displacement

41 b) Plot a velocity-versus-clock reading graph for the motion represented in the position vs. time graph. 7.3 Represent and Reason 6.0 Velocity vs. Time Velocity (m/s) time (s) PUM Kinematics Lesson 7: Using Velocity-versus-Time Graphs to find Displacement 41

42 The figure above shows a velocity vs. time graph that represents the motion of a person moving along a straight hiking path. The positive direction of the coordinate axis is toward the south. a) Describe the motion of the hiker in words. b) Use the graph to determine how far the hiker moved from the clock reading of 10 s to the clock reading of 25 s. Explain. c) Use the graph to estimate his distance traveled for the time interval 40 s to 70 s. d) What is the average speed of the hiker? Explain. 7.4 Evaluate The following graphs represent the motions of two bicyclists. With which of the statements about the motions do you agree? Explain your choice. v ( m/s) x (m) t (s) t (s) a) Bike A started moving at constant positive velocity. b) Bike B climbed over a flat hill. c) Bike A stopped twice during the trip. d) Bike B stopped twice during the trip. e) The last part of the trip bike A was not moving. f) The last part of the trip bike B was moving at constant speed in the negative direction. g) The last part of the trip bike A was moving at constant speed in the negative direction. h) When we started observing Bike B it was moving at constant positive velocity. i) When we started observing bike A it was moving at increasing velocity in the positive direction, then it reached some constant velocity (positive) and continued moving for a while, then its velocity started decreasing and it some point it became zero. The it continued to increased in the negative direction until it reached some new velocity which it maintained for a while. 42 PUM Kinematics Lesson 7: Using Velocity-versus-Time Graphs to find Displacement

43 j) When we started observing bike B it was moving at constant velocity in the positive direction, then it stopped for a while, then it started going back to the origin and then in the negative direction. Finally it stopped. Here s an Idea! Now that you are familiar with different kinds of graphs position versus time, displacement versus time and velocity or speed versus time, you might confuse them when solving problems. To avoid confusion, every time you start working with a graph, slow down for a moment and say: Hello Mister Graph. My name is (say your name here). I see that you are a velocity versus time graph (or a position versus time depending on what graph it is)! After you greet a graph like this, you will certainly avoid the confusion. And if it sounds silly, it s ok. 7.5 Represent and Reason Homework You are driving home from the University of Delaware after a college visit at 65 miles per hour (mph) for 130 miles. It beings to rain and you slow to 55 mph. You arrive home after driving for 3 hours and 20 minutes. a) Diagram and label all the pertinent information for this trip. b) Plot a position vs. time graph for this motion. (Do not forget to say Hello Mister Graph! ) c) Plot a velocity vs. time graph for the trip. (Do not forget to say Hello Mister Graph! ) What is the average velocity for the total trip? What is the average speed for the total trip? 7.6 Evaluate Examine the graphs below. Then choose the statements with which you diagree. Explain your reasons. Then explain why someone would choose those wrong answers as correct. A B C D 30 x, m v, m/s 30 v, m/s t, s t, s t, s a) The first two graphs (A and B) provide the same information. b) The second two graphs (C and D) provide the same information. x, m t, s PUM Kinematics Lesson 7: Using Velocity-versus-Time Graphs to find Displacement 43

44 c) Object A traveled 60 meters in 3 seconds from the location it was at the 0 clock reading. d) Object B traveled 40 meters in 2 seconds in the positive direction. e) Object C was not moving during the experiment. f) Objects A and D were not moving during the experiment. g) Object C was moving in the negative direction at the speed of 20 m/s. h) Object D was moving in the negative direction at the speed of 20 m/s. i) Object C was moving in the negative direction at the speed of (-20 m/s). j) Object D traveled 40 m in 2 seconds in the negative direction. 7.7 Pose your own problem Make up a problem to solve which one needs to know how to calculate displacement of an object from a velocity versus time graph. Solve the problem and make a list of difficulties that your friends might have with it. (Hint: to start, you can modify one of the problems in the lesson.) Reflect: What did you learn in this lesson? What was easy? What was difficult? What could you have done differently to ease the pain? 44 PUM Kinematics Lesson 7: Using Velocity-versus-Time Graphs to find Displacement

45 8.1 Design an Experiment Lesson 8: Lab: Will the Cars Ever Meet? The goals of this experiment are: to apply kinematics equations to describe real-life phenomena; to become aware of assumptions that one makes when applying mathematical equations to a real situation; to understand that one cannot avoid experimental uncertainties with particular equipment. In this experiment, you will: determine the kind of motion two cars have, and use kinematics equations to predict the location of where the two cars will meet if they start at given locations and move toward each other. Equipment: Two battery-operated cars, meter stick, stopwatch, sugar packets or any kind of markers for the cars locations Setting up the experiment: First, experiment individually with the cars to determine whether each car moves with constant speed or changing speed and then determine the speed of each car. You can use the methods you learned in the first lab to record positions of the cars every second. Use the following steps: a) Draw a clearly labeled diagram of your experimental setup. b) List the physical quantities you will measure and how you will measure them. c) Perform the experiment. Record the data in appropriate formats, such as a dot diagram, a table, and a graph. d) What is the uncertainty in your measurements for clock reading and position? Which uncertainty is the largest in your experiments? e) How can you represent the uncertainty on your graph? Does it change the way that you would draw the trend lines? f) Find the speed of each car. Did You Know? Experimental uncertainty When we collect data there are two kinds of uncertainty. First, instrumental uncertainty is due to the fact that the scales of the instruments have divisions. We cannot measure anything more precisely than half of the smallest division. We will learn more about this PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet? 45

46 uncertainty later in the lesson as in this particular experiment another type of uncertainty is more important. It is called random uncertainty which is due to the fact that sometimes it is difficult to collect data consistently. For example, when you measure how much time it takes for one of your cars to move 2.0 m and obtain a particular value for the time interval, there is little guarantee that next time you repeat the same experiment you will get exactly the same value. It is due to the fact that you cannot release the car the same way in both experiments, or that the car encounters a bump on the floor, or some other reason. To find the random uncertainty, you need to measure the same value several times (at least three) let s say you measured n 1 ; n 2 ; n 3. Next you find the average of three numbers n = n + n + n Then you find how far in terms of the magnitude each measurement is 3 from the average: n 1 = n n 1 ; n 2 = n n 2 ; n 3 = n n 3 and finally choose the largest of the three. This is the largest uncertainty in the value you are trying to determine. Now you can say that the value of your measurement is where n largest is the largest random uncertainty of your measurement. Sometimes it is more useful to express the uncertainty not as an absolute number but as a percent uncertainty ( n largest 100%). n 8.2 Design an Experiment This time you need to predict where the cars would meet if they were placed 3 m apart initially and started at the same time. The steps outlined below will help you with reasoning through this prediction. a) Think of how you can represent the motion of each car with the equation x(t). Remember that the cars start at two different locations and move in different directions! b) Use the equations that you wrote to predict where the cars will meet if released simultaneously. How certain are you in the location? Did you know? Instrumental Uncertainty To express your prediction as a range and not as a single number: x = x 1 ± the value of the uncertainty, you need to think of how you know what the uncertainty is. You have not performed the experiments, so there random uncertainty is not known. However, we as we talked before, every instrument has the uncertainty due to the size of its smallest division. For the ruler it might be 0.5 mm. If you are using more than one instrument for example a watch and a ruler, then the instrument that brings in the biggest uncertainty will determine your final instrumental uncertainty. For example the watch has an uncertainty of 0.5 seconds and a ruler has an uncertainty of 0.5 mm. How can you compare mm to seconds? To answer this question we will use the weak link rule. This is how it works. We estimate the relative uncertainty that each instrument gives us this is the half of the smallest 46 PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet?

47 division divided by the smallest measurement you make with this instrument and compare to the relative uncertainty due to the second instrument. The instrument with the largest relative uncertainty will determine the instrumental uncertainty of your experiment. For example you measure 3 m length with the ruler that has 1 mm divisions, thus your relative uncertainty for the ruler is 0.5 mm/3 m = (0.5 x 10-3 m)/ (3m) = 0.2 x 10-3 (or 0.02% of the measurement); you also measure 10 seconds with your watch, the uncertainty is 0.5 s / 10 s = 0.05 (or 5 % of the measurement). This is much larger than the 0.2 x 10-3 relative uncertainty by the ruler. Thus if you use both instruments, you need to report your results (or predictions) within the interval of 0.05 times (5%) the measured value: for example: instead of 3 meters, we will need to write 3 m ± (0.05 x 3 m) = 3 m ± 0.15 m. Actually, you can use the weak link rule to evaluate any uncertainties in your result: first you evaluate random uncertainty as the percent of the result, then the instrumental whichever is larger will give you the estimate of the uncertainty your result has. c) When you make the prediction, think of the assumptions that you are using. One of them is that the cars move at constant speed right from the start of the measurement. How will this assumption affect the meeting location? What are other assumptions you are making? d) Show your prediction to your instructor and then try the experiment. Record the outcome. Make sure that your experiment allows you to evaluate random uncertainty and compare it to the instrumental. e) Reconcile any differences in your prediction and the outcome of the experiment. Be specific. Can you explain any differences because of experimental uncertainties or assumptions that might not be valid? 8.3 Write a lab report Homework Write a scientific report about the two experiments that you performed. The report should describe everything you did so that a person who did not do the experiments could repeat them and get the same results. Make sure that you justify your judgment about the outcome of the experiment. Use the rubrics to improve your lab report. Make sure you study each of the abilities and decide whether you can improve your report based on the rubrics. PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet? 47

48 Ability to collect and analyze experimental data Scientific Ability Missing An attempt Is able to identify sources of experimental uncertainty. Is able to specifically evaluate how identified experimental uncertainties may affect the data. Is able to record and represent data in a meaningful way. Is able to identify the assumptions made in using the mathematical procedure. Is able to specifically determine the ways in which assumptions might affect the results. No attempt is made to identify experimental uncertainties. No attempt is made to evaluate experimental uncertainties. Data are either absent or incomprehens ible. No attempt is made to identify any assumptions. No attempt is made to determine the effects of assumptions. An attempt is made to identify experimental uncertainties but most are missing, described vaguely, or incorrect. An attempt is made to evaluate experimental uncertainties but most are missing, described vaguely, or incorrect. Or the final result does not take the uncertainty into account. Some important data are absent or incomprehensible. An attempt is made to identify assumptions, but the assumptions are irrelevant or incorrect for the situation. The effects of assumptions are mentioned but are described vaguely. Needs some improvement Most experimental uncertainties are correctly identified but the source of the biggest uncertainty is not specified. The final result does take the identified uncertainties into account but is not correctly evaluated. All important data are present but recorded in a way that requires some effort to comprehend. Relevant assumptions are identified but are not significant for solving the problem. The effects of assumptions are determined, but no attempt is made to validate them. Acceptable All experimental uncertainties are correctly identified and the source of the biggest uncertainty is specified. The experimental uncertainty of the final result is correctly evaluated; the final result is written within the margin of uncertainty. All important data are present, organized, and recorded clearly. All relevant assumptions are correctly identified. The effects of the assumptions are determined and the assumptions are validated. 8.4 Represent and Reason a) Several motions are described mathematically below. All quantities are in SI units (position in meters and time in seconds). Represent these motions in words and graphically as x(t) and v(t). Draw the position and velocity versus time graphs without creating a data table; use the information about the slope and the y-intercept to draw the graphs. Make sure that you draw all the x(t) graphs using the same set of axes and all the v(t) graphs using the same set of axes (make sure you say Hello Mister Graph to each of the graphs when you start drawing them).. x = +3t + 5 x = (-3t) + 5 x = (-3t) +( 5) x = 3t +( 5) x = 3t x = -3t 48 PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet?

49 b) If you choose a reference frame in which a toy car is traveling at a constant speed and then you decide to change the reference frame to a different one, will the car still be traveling at constant speed? Explain. c) The motion of two objects are described by two x(t) functions: x = (-12) + 3 t and x* = +24 +( 7 t). All quantities are in SI units. Describe the motion of the two objects in words and with graphs of x(t). Use the equations to determine where and when the objects will meet. What assumptions did you make? 8.5 Represent and Reason Consider the graphs below for objects A and B. Graph 1 Graph 2 A A Position (m) B Position (m) B Time (s) 6.0 sec Time (s) 6.0 s How does the motion of the object A in graph 1 compare to that of A in graph 2? a) Say Hello Mister Graph to the graphs. What do graphs represent? b) How does the motion of object B in graph 1 compare to the motion of object B in graph 2? c) Which object has the smaller speed in graph 2? Explain. d) Describe what is happening at the point where the function of object A intersects that of object B. e) Which object traveled a greater distance during the first 6 seconds in graph 1? Explain. f) Write a function x(t) for all four objects. PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet? 49

50 8.6 Represent and Reason A New Jersey Transit train leaves Bay Head and heads north to Secaucus. Another train leaves Secaucus and heads South to Bay head, both trains are express trains (which means they do not stop). If the northbound train has an average velocity of 35 mi/hr and the southbound has an average velocity of 52 mi/hr. The distance between the two stations is 60 miles, predict where the two trains will meet. Be sure you address the assumptions you made in order to solve this problem. 8.7 Represent and Reason A student starts walking at 5 ft/s in a corridor A and is 20 ft away from the intersection of corridors A and B. A second student starts at the same time running at 8 ft/s in corridor B and is 32 feet away from the intersection. a) Create a dot diagram for the problem. b) Graph the motion of the two students on a position-versus-time graph c) Represent the motion of each student with a function. d) How many different functions can you write for motion of each student? What will be different and what will be the same? e) Will the two students collide? Show your work and explain your reasoning. 8.8 Reason Examine the graph. What is the object s average speed during the first 4 seconds of its trip? I) 17.5 m/s; II) 4.4 m/s (17.5/4); III) 8.8 m/s (17.5/2) x, m/s t, s 50 PUM Kinematics Lesson 8: Lab: Will the Cars Ever Meet?

51 9.1 Describe Lesson 9: Motion Diagrams During art class you made a clay figure of a person and the art teacher was very impressed by your work. She even displayed your work for the whole class. However throughout the week you noticed that people were changing your artwork! Day 1 Day 2 Day 3 Day 4 Day 5 For each day, determine if you artwork was stretched up, squished down, or unchanged. Then draw an arrow to represent the direction of change. Make sure your arrow shows the amount of stretching or squishing. Original to Day 1: Day 1 to Day 2: Day 2 to Day 3: Day 3 to Day 4: Day 4 to Day 5: Here s An Idea! This activity may not have seemed like physics but it is design to help you understand change arrows. This activity serves as an analogy for you to refer back to when you re having trouble. Scientist often will make analogies for complex systems in order to better understand what may be occurring. PUM Kinematics Lesson 9: Motion Diagrams 51

52 9.2 Represent and Reason Use a small ball and a long tilted ramp at a very small angle. Let the ball roll down the ramp. a) Draw a dot diagram that shows the velocity during each time interval, direction of motion, and a directional axis. Need Some Help? What you need to draw is something physicists call, a motion diagram. It is a sophisticated replacement for a dot diagram that conveys more information about a situation. If you are new to this representation, you may want to list or label the important features so that you are sure to include these when you draw one for yourself. Example: v 1 v 2 v 3 v 4 b) What does the length of the arrow tell you about the motion of the object? c) What two ways is direction indicated on the representation? Why are both necessary? d) What does the length of each arrow tell you about the motion of the car at a particular dot or position? e) What does the length of each arrow tell you about the motion of the ball at a particular time? f) With a classmate, act out the motion represented in the motion diagram. g) If you were to make a position versus time graph for this motion how would the trend line look? Why does this make sense? 9.3 Observe and Represent v 1 v 2 v 3 v 4 52 PUM Kinematics Lesson 9: Motion Diagrams

53 In this activity, we re going to examine the motion diagram provided in the previous activity. a) Think about the previous activity. What must you do to velocity arrow one to get velocity arrow two? What direction? How much? b) Answer the questions in part (a) for arrows 2 and 3 as well as arrows 3 and 4. Is there any difference? Did you know? The arrows that you drew to show to difference between velocity arrows are called v arrows or change in velocity arrows. We can line them up and compare the size of the arrows in order to determine the change. These arrows tell us the direction and the magnitude (size) of the change. It s just like the direction and amount of squishing/stretching from the previous activity! A complete motion diagram includes v arrows. Example: v 1 v 2 v 1 v 2 The second v arrow is longer than the first so it s a stretched to the right (positive direction) and shows speeding up. c) What does the v arrow tell you about the motion of the object? Explain. d) Revisit the dot picture you drew for the ball rolling down an incline. Create a complete motion diagram for the motion of the ball. 9.4 Reason and Explain Describe the motion represented in this motion diagram. v 4 v 3 v 2 v 1 a) Is the object speeding up, slowing down, or moving at constant velocity? How do you know? b) What direction should the v arrow be pointing? Explain how you determined this. c) Is the change in motion ( v arrows) in the same direction that the object is moving (v arrows)? d) Make a rule for speeding up and slowing down by comparing motion (velocity arrows) to change in motion ( v arrows) PUM Kinematics Lesson 9: Motion Diagrams 53

54 Did You Know? Constructing a Motion Diagram: Here is a tool to help you learn how to construct motion diagrams. Motion diagrams provide a concrete way to represent motion. Motion diagrams help you to represent, visualize, and analyze motion. They are especially useful for checking the quantitative math descriptions of motion that you will learn later. Draw dots to represent the position of the object for equal time intervals. Point arrows in the direction of motion and use the relative lengths to indicate how fast the object is moving between the points. v 1 v 2 v 3 Draw arrow to indicate how the arrows are changing. Draw the arrows thicker than the arrows. 9.5 Represent and Reason Homework a) Make up a story about the motion of some object and represent it with a motion diagram. b) Draw motion diagrams for: a car starting to speed up from rest next to at a street light, for the car coasting along a street with a constant speed passing the street light along the way, and for the car slowing down to a stop next to a street light. c) Draw a picture of the three situations above and then sketch a position-versus-time graph for each. Assume that the streetlight is the origin. d) Draw a picture of the three situations above and then sketch a position-versus-time graph for each. Assume that the streetlight is the origin. 9.6 Represent and Reason The table below describes experiments for the motion of a ball. For each experiment, something happens to the ball and its motion changes. Visualize every experiment and if you have a ball perform it! Use the information in table and your experiments/imagination to fill in the blanks in the table below. 54 PUM Kinematics Lesson 9: Motion Diagrams

55 Initial motion v Final motion Motion diagram to match a) b) c) d) e) f) Not moving Moving in y direction Moving in x direction Moving in +x direction x direction Moving faster in y direction Not moving in x direction x direction Not moving in x direction +x direction +y direction Moving in y direction 9.7 Represent and Reason The table below describes the velocity of a ball for different experiments. For each experiment, something happens to the ball and its velocity changes. Use the information in table to fill in the blanks in the table below. Initial velocity, v i v Final velocity, v f Motion diagram to match a) b) c) d) e) Reflect: What did you learn in this lesson? How is motion diagram similar to a dot diagram? How is it different? PUM Kinematics Lesson 9: Motion Diagrams 55

56 10.1 Test your Idea Lesson 10A: Free Falling? In front of the class, drop an object such as a tennis ball. Observe the motion of the ball as it falls. a) As a class, discuss the motion of the falling object. What do the students think about the motion and why? b) Design an experiment that will allow you to use the derived kinematics equations to predict the time interval if you dropped the object from the bleachers (or some other high place). In designing your experiment what assumptions are you making about the motion? c) Before experimenting, discuss the outcomes. If the outcome is the same is as the predicted time, what does this mean about the kinematics expression and the assumption(s)? If the outcome is different than the predicted time, what does this mean about the kinematics expression and the assumption(s)? Scientific ability Is able to distinguish between a hypothesis and a prediction Rubric to self-assess your prediction Missing An attempt Needs some improvement No prediction is A prediction is A prediction is made made. The made, but it is and is distinct from the experiment is not identical to the hypothesis but does not treated as a testing hypothesis. describe the outcome of experiment. the designed experiment. Acceptable A prediction is made, is distinct from the hypothesis, and describes the outcome of the designed experiment Reflect Homework Reflect on the last nine lessons while considering the mathematical model x(t) = x 0 + v(t). Summarize the assumptions made while using this model Observe and Find a Pattern Go to on the video website and collect position versus time data for the falling object. Use the data to draw a dot diagram for the falling ball. What can you say about its motion based on the diagram? 56 PUM Kinematics Lesson 10A: Free Falling?

57 Lesson 10B*: Review of Graphical Representations 10B*.1 Hypothesize Let s review position versus time graphs. Use the ideas developed from the previous lessons to help you develop the following rules. a) What does constant pace motion look like on a position versus time graph? b) What does speeding up motion look like on a position versus time graph? c) What does slowing down motion look like on a position versus time graph? 10B*.2 Test Your Idea Use your newly modified hypotheses from the previous activity to predict how you d have to move so that a motion detector creates position versus time graphs that match the previous graphs. Explain how your prediction compares to the outcome. 10B*.3 Reason Examine the graphs below and then answer each of the questions below by recording the associated letters on the line provided. The units for time are seconds. a) Which graphs represent objects moving at constant pace? b) Which graphs represents objects speeding up? c) Which graphs represent objects slowing down? d) Which graphs represent an object moving in the negative direction? e) Do any of the graph show an object that is not in motion? How do you know? Can we consider this a constant pace? PUM Kinematics Lesson 10B*: Review of Graphical Representations 57

58 a b c d e f 58 PUM Kinematics Lesson 10B*: Review of Graphical Representations

59 I. Speeding Up Index Lesson 11: Part I Speeding index Today you will look at another kind of index. Your task is to come up with a speeding up index for cars. You will see pictures of several cars on the next page, and you need to come up with one number to stand for each car s speeding up. There is no watch or clock to tell you how long each car has been going. However, all the cars drip oil once a second. (They are not very good cars!) The speedometer reading tells you how fast the car is going when the oil drips. Oil drops This task is a little harder than before. A company always makes its cars speed up in the same way. We will not tell you how many companies there are. You have to decide which cars are from the same company. They may look different! To review: (1) Make a speeding up index for each car. A bigger index means a car speeds up more. (2) Decide how many companies there are. (3) To show the cars that are from the same company, draw a line that connects the cars. Reasoning Questions: 1. Use everyday language to describe the specific information that the speeding up index tells you about the car s motion. 2. How many car companies are there? How do you know? 3. It can be easy to confuse fast with speeding up quickly, or slow with speeding up slowly. What is the difference? Use the following questions to help sort this out: a) Which car is the fastest? Explain what fastest means. b) How long will it take the fastest car to speed up by 75 mph? c) Which car is the slowest? Explain what slowest means. d) How fast will the slowest car be going 10s after it starts? PUM Kinematics Lesson 11: Part I Speeding index 59

60 . e) Compare fast to speeding up quickly and slow to speeding up slowly. Can a car be going fast and speed up slowly? Can a car be going slow and speed up quickly? Explain. Car A 5 mph 25 mph 45 mph Car B 0 mph 9 mph 15 mph Car C 5 mph 50mph 10 mph 55 mph 5 mph 50 Car D 4 mph 14 mph 24 mph 34 mph Car E 57mph 12 mph 72 mph PUM Kinematics Lesson 11: Part I Speeding index

61 Car A Car B Car C Car D Car E 5 mph 0 mph 5 mph 4 mph 12 mph 15 mph 3 mph 6 mph 20 mph 14 mph 27 mph 25 mph 9 mph 24 mph 42 mph 35 mph 35 mph 34 mph 57 mph 50 mph PUM Kinematics Lesson 11: Part I Speeding index 61

62 Lesson 11 Part II: Motion of a Falling Object 11.1 Observe and find a pattern Vernier Software Video or tinyurl.com/3akwvjy Use the video of the falling ball to collect position versus time data. Notice that for a few frames the data are missing, it should not affect your results. a) Describe the motion of the ball. b) Decide on an origin and measure the position of the ball at the beginning of each time interval. How did you decide how to measure the position? c) What is the instrumental uncertainty in your measurements? In case you cannot access the videos, a few data points are provided in the table below Clock Reading (s) Position (m) Need Some Help? More about the instrumental uncertainty For example, you are measuring the length of the table to be 2m with a ruler that has centimeter divisions. The half of the smallest division is 0.5 cm. You can write your measurement as l = m ± 0.005m. 0.5 cm constitutes 0.25% of 2 m, thus we can also write the measurement as l = m ± 0.25%. The latter way of writing the instrumental uncertainty explains why using the same instrument to make larger measurements gives you a more precise result than making smaller measurements (for example 0.5 cm is 0.25% of 2 m but it is 10% of 5 cm, thus when you measure the distance of 5 cm with this ruler, you have 10% uncertainty). d) Record the position and time data for the ball in a table. Can you see any patterns? Explain. 62 PUM Kinematics Lesson 11 Part II: Motion of a Falling Object

63 e) Plot a position versus time graph for this object. What type of function does the trend line resemble? Does this graph represent an object traveling with constant velocity? How do you know? 11.2 Represent and Reason a) Now we will examine graphically how the velocity changes. Calculate the average velocity for the ball for each time interval and complete the table that follows. What is the uncertainty for the velocities? (You may use the table of data in activity 11.1 or collect your own data) Time interval t =t n t n-1 Displacement x = x n x n-1 Average clock reading (t n + t n-1 )/2 Average velocity b) What patterns do you notice in the table, what do these patterns indicate about the motion of the falling ball? c) Plot an average velocity versus time graph. Write a function for how the speed changes with time, v(t). d) What is the meaning of the slope of this line? Think about the physical meaning of the slope of the line and name it. e) Discuss how the equation would change if the ball were slowing down instead of speeding up. Discuss how your equation would change if you the ball were initially thrown upwards. f) As we have seen, the notion of a rate of change is an important mathematical and scientific idea. In this case, the rate at which the velocity of an object changes is referred to as the acceleration of the object. The rate at which the distance traveled by an object changes is called the velocity of the object. Did you know? Acceleration of an object moving at constant acceleration is the slope of the velocity versus time graph and is equal to the change in velocity of the object divided by the time interval during which this change in velocity occurred. When the object is moving at constant acceleration this ratio is the same for any time interval PUM Kinematics Lesson 11 Part II: Motion of a Falling Object 63

64 a = v 2 v 1 t 2 t 1 where v 2 v 1 ( v) is any change in position during the corresponding time interval t 2 t 1 ( t ). The unit for acceleration is m/s/s or m/s Practice Imagine a car moving at a speed of 10 m/s. It starts speeding up at an acceleration of 2 m/s/s. Do not use any formulas, just your understanding of acceleration, to decide what the car s speed will be after 2 second; 4 seconds, 6 seconds Practice Imagine a car moving at a speed of 10 m/s. It starts slowing down at an acceleration of 2 m/s/s. Do not use any formulas, just your understanding of acceleration, to decide what the car s speed will be after 2 second; 4 seconds, 6 seconds Reason a) You know that velocity has magnitude (called speed) and direction. What about acceleration? If you think it does, how can you argue your point? If you think it does not, how do you argue your point? Find a student in class who disagrees with you and try to discus this issue with her/him. b) How is acceleration related to the velocity change arrow on the motion diagram? Explain Represent and reason The motion diagrams in the illustrations below represent the motion of different objects modeled as point-like objects. The arrows are velocity arrows. 0 II y I x I II I I x 0 II 0 A different coordinate axis is provided in each situation. An open circle indicates the location of interest. 1. Draw velocity change arrows on each diagram above. 64 PUM Kinematics Lesson 11 Part II: Motion of a Falling Object

65 2. Fill in the table that follows. Be sure to make your choices relative to the coordinate axis shown with each motion diagram. Describe the motion in words. Determine the sign (+, 0, or ) of the position. Determine the sign (+, 0, or ) of the velocity. Determine the sign (+, 0, or ) of the acceleration. a) Location I: Location I: Location I: b) Location I: Location I: Location I: Location II: Location II: Location II: c) Location I: Location I: Location I: Location II: Location II: Location II: Location III: Location III: Location III: 11.7 Reason What does it mean that the velocity of a bicyclist is m/s? What does it mean if this bicyclist is accelerating at 2 m/s/s? at -2 m/s/s? 11.8 Evaluate Tara says that if an object has negative acceleration, it is slowing down. Do you agree or disagree? How can you argue your point of view? 11.9 Represent and Reason The graph below represents the motion of a person riding a moped over a period of time as seen by the observer on the ground. South is denoted by positive numbers. a) What kind of information does this graph represent? (Do not forget to say Hello Mister Graph! ) b) What was the speed of the moped at the beginning of observations? What was the velocity? What was the speed at 25 s clock reading? What was the velocity? c) What happened to the speed between 20 and 44 seconds? Between 44 and 65 seconds? When was the moped at rest? When did it climb a hill? PUM Kinematics Lesson 11 Part II: Motion of a Falling Object 65

66 d) What accelerations did the moped have during the trip? Provide numbers (include the signs). e) Compare the distance traveled during the first 20 seconds and the second 20 seconds. Which one is bigger? How do you know? f) For the clock reading of 80 seconds choose the answer that you think is correct: I) The graphs is flat, so the moped must have stopped.; II) The moped s speed was 0.15 m/s (12 divided by 80); III) The moped s seed was 0.34 m/s (12 divided by 35); IV) the speed was 12 m/s (the graph shows velocity vs time) Represent and Reason Homework Study the following three graphs taken from actual laboratory data. Determine the acceleration for the motion represented on each graph. What does it mean if the acceleration is positive? Negative? Write the functions v(t) for the following graphs (on the next page). 66 PUM Kinematics Lesson 11 Part II: Motion of a Falling Object

67 a) Describe a situation that will match each graph. b) Compare the velocity versus time graphs for objects that move at constant velocity and objects that move with changing velocity. c) What can you say about the acceleratio n of the moving objects in the graphs? Explain. In the following problems use the problem solving strategy! Practice On a bumper car ride, friends smash their cars into each other (head-on) and each has a speed change of 3.2 m/s. If the magnitudes of accelerations of each car during the collision averaged 28 m/s 2, determine the time interval needed to stop for each car while colliding. Specify your reference frame Practice PUM Kinematics Lesson 11 Part II: Motion of a Falling Object 67

68 A bus leaves an intersection accelerating at +2.0 m/s 2. What is the speed of the bus after 5.0 s? What assumption did you make? Explain Practice What was the speed of a jogger if she needed 2 seconds to stop at the acceleration of 3 m/s/s. Is the answer realistic? Estimate Estimate the acceleration of a car one of your family members drives. How did you make the estimation? How many significant digits do you have in your result? Practice A jogger is running at +4.0 m/s when a bus passes her. The bus is accelerating from m/s to m/s in 8.0 s. The jogger speeds up at the same acceleration. What can you determine about the jogger s motion using these data? Represent and Reason A person on a motorcycle facing South is initially at rest. The motor cycle speeds up from rest to 15 m/s in three seconds. For the next 8 seconds, the motorcycle travels at the same speed. Over the next 12 seconds the motorcycle slows to rest uniformly. Plot a acceleration-versus clock reading graph for the motor cycle. What assumptions did you make about the motorcycle and its motion? Represent and Reason The figure below shows a velocity-versus-clock reading graph that represents the motion of an object, modeled as a point particle (we are not interested in the motion of the feet, head, etc.), moving along a straight path. The positive direction of the position axis is toward the North. 68 PUM Kinematics Lesson 11 Part II: Motion of a Falling Object

69 a) Describe the motion of the object in words. b) When does the object travel at constant velocity? Explain how you know. c) When does the object have a velocity equal to zero? Explain how you know. d) What is the average acceleration for the first three seconds? e) What is the average acceleration for the time interval seconds? f) Plot an acceleration-versus-clock reading graph for the motion of the object. g) Examine 4.5 seconds, what can you say about the object s acceleration and velocity at that clock reading? Observe and represent The false-color image on the next page shows the Gangotri Glacier, situated in the Uttarkashi District of Garhwal in the Himalayas. Currently 30.2 km long and between 0.5 and 2.5 km wide, Gangotri Glacier is one of the largest in the Himalayas. Gangotri has been receding since Use the satellite image to estimate the speed at which the glacier is receding. Describe your procedure and collect all data from the image. PUM Kinematics Lesson 11 Part II: Motion of a Falling Object 69

70 Reflect: What did you learn during this lesson? How acceleration is similar to velocity? How is it different? Why do you need both? 70 PUM Kinematics Lesson 11 Part II: Motion of a Falling Object

71 Lesson 12: Position of an Accelerating Object as a Function of Time 12.1 Hypothesize (Derive a Mathematical Model) Recall the initial position and clock reading data from the previous lab. When considering the motion of the falling ball, how is position related (mathematically) to time? Use the method developed in lesson 11 to find a relationship between the displacement of an object (moving with constant acceleration during some time interval), its velocity at the beginning of this interval, its acceleration, and the length of the time interval. Start by drawing a velocity-versus-clock reading graph and examining the area under the graph line. Express the relationship in terms of v 0, a, and t. Use the expression for the displacement to write the function x(t) for the object moving at constant acceleration. Write the function in terms of x 0, v 0, a, and t Test Your Idea with Phet Simulations Go to and click on Run Now! Click on the Charts tab in the upper left corner of the window. You should see a screen like the one shown. There is a man at top of the simulation who can move 10 m in either direction from the origin. The simulation also includes axes of position, velocity and acceleration graphs that will reflect his motion. Since you are not going to use the acceleration graph right away, you can close it by clicking on the small window in the upper right hand corner of each section. To eliminate the walls, click on special features, then click on free range. Use the hypothesized mathematical model in activity 12.1 to predict the position. Scenario 1: The man s initial position is at the tree where he is initially at rest. He has an acceleration of 0.75 m/s/s to the right. a) Write an expression for the man s position as a function of time. b) Create a position vs. time and velocity vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the location of the moving man after 5 seconds. Show your work PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time 71

72 e) Perform the experiment by entering given quantities in the respective simulation boxes and click Go! Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. Scenario 2: The man is walking initially at 0.75 m/s towards his home starting from the position of 7m to the left of the origin. At this point, he begins to increase his velocity at a rate of 0.2 m/s every second. a) Write an expression for the man s position as a function of time. b) Create a position vs. time and velocity vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the time when he arrives at the origin. e) Perform the experiment through the simulation. Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. Scenario 3: The man starts at the 5m mark by the house and is walking towards the 1.0 m/s towards the tree. He is accelerating towards the tree at 0.5 m/s/s a) Write an expression for the man s position as a function of time. b) Create a position vs. time and velocity vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? d) Predict the position when the man is moving at a speed of 5 m/s. e) Perform the experiment through the simulation. Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion. Scenario 4: The man starts at the house running at 7.0 m/s towards the tree. He is slowing up at 1.0m/s/s. a) Write an expression for the man s position as a function of time. b) Create a position vs. time and velocity vs. time graph for this function. c) Before you continue with the simulation, check for consistencies between the written description, function and graph for the man. How do you know they are consistent? 72 PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time

73 d) Predict the position of the man s final position when he comes to rest. e) Perform the experiment through the simulation. Compare your predicted value to the outcome of the testing experiment. Do they agree or disagree? If they disagree, revise your mathematical model of the moving man s motion Practice Velocity of a motorcycle changes according to the graph below. a) What do the slopes of the line segments on the graph tell you? When does the motorcycle have a positive acceleration, zero acceleration, negative acceleration? b) What is the distance traveled during the first 20 seconds? From 20 to 45 seconds? From 45 to 65 seconds? From 65 to 85 seconds? From 85 seconds to the end of the recorded trip? c) Determine the total path length traveled and the displacement of the motorcycle. Make sure you use the Problem Solving Strategy for the problems below Practice A bus leaves an intersection accelerating at +2.0 m/s 2 from rest (think what the term rest means). Where is the bus after 5.0 s? 12.5 Practice A bicyclist slowed down from 8 m/s to 2 m/s in 3 seconds. What was the acceleration of the bicycle? How far did it move during this process? 12.6 Reason The motion of a car can be described by the following function. All quantities are in SI units x(t) =14t + 3t 2 PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time 73

74 Explain the meaning of each number. Describe the motion in words, with a motion diagram, and with a picture with a reference frame. a) What would the v(t) expression look like? b) Determine the position and the velocity of the car after 5 seconds. Another car s motion is describe by the following equation: x(t) = (-12) + 7t + ( 0.4t 2 ) c) How does this motion compare with the previous car? Repeat (a) and (b) for this object. d) Act out each motion. To act out, have two students to represent each moving objects. Let classmates give those students directions on how to move Compare and Contrast Jim says: We learned so many different words: constant velocity, zero velocity, constant acceleration and zero acceleration. I do not understand the difference between them, all sound like motion to me. Do you feel similar to Jim? If you do, it is normal. To help yourself navigate through the new ideas, work through the following exercises. Draw a motion diagram for each scenario to help you construct each situation. a) Describe a situation when an object moves with an acceleration equal to zero and a velocity that is a non-zero negative number. b) Describe a situation when an object moves with an constant positive acceleration and velocity is positive number. c) Describe a situation when an object travels with a constant negative acceleration and an positive velocity. d) Describe a situation when an object travels with a constant negative acceleration and a negative velocity Reason The driver of a car moving east a speed v o sees a red light in front of him and hits the brakes after a short reaction. The car slows down at a rate of a 1,2. A typical reaction time is 0.8 seconds. The situation is represented in the picture. a) Where is the origin of the reference frame? b) What information given in the problem is missing from the illustration? Add it to the illustration. c) What assumptions are made in Part I and Part II? How do these assumptions affect the mathematical expressions that you can use in each part? 74 PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time

75 Homework 12.9 Represent and Reason A stoplight turns yellow when you are 20 m from the edge of the intersection. Your car is traveling at 12 m/s; after you hit the brakes, the car's speed decreases at a rate of 6.0 m/s each second. (Ignore the reaction time needed to bring your foot from the floor to the brake pedal.) a) Sketch the situation. Decide where the origin of the coordinate system is and what direction is positive. b) Draw a motion diagram. c) Draw an x(t) graph. d) Draw a v(t) graph. e) Write an expression for x(t) and v(t). f) Use the expressions above to determine as many unknowns as you can Represent and Reason A bus moving at 26 m/s (t = 0) slows at rate of 3.5 m/s each second. Sketch the situation. Decide where the origin of the coordinate system is and what direction is positive. a) Draw a motion diagram. b) Use the expressions derived in this lesson and previous lessons to determine as many unknowns as you can Reason and Represent An object moves horizontally. The equations below represent its motion mathematically. Describe the actual motion that these two equations together might describe. a. v = +20 m/s + ( 2 m/s 2 )t b. x = 200 m + (+20 m/s)t ( 2 m/s2 )t 2 a) Describe the motion in words and sketch the process represented in the two mathematical expressions above. Act it out. b) Draw a motion diagram c) Draw a position-versus-clock reading graph and a velocity-versus-clock reading graph PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time 75

76 d) Determine when and where the object will stop Reason and Represent A remote control car runs down a driveway at an initial speed of 6.0 m/s for 8.0 sec, then uniformly increases its speed to 9.75 m/s in 5.0 sec. a) Sketch the situation, label all knowns and unknowns. Decide where the origin of the coordinate system is and what direction is positive. b) Draw a motion diagram. c) Draw a v(t) graph. d) Use the expressions in this and previous lessons to determine as many unknowns as you can Regular problem Examine the graph below. a) Describe a real life situation that this graph could represent, be sure to include all the information on the graph and any extra in your situation. b) Determine two unknown physical quantities (one of them should be in the units of meters). c) If the object was moving at a constant speed equal to the speed of the object on the graph at t = 0, what would be the distance it traveled in 6 seconds? How does it compare to the distance the object on the graph traveled in the same time interval? Does the answer make sense to you? 76 PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time

77 Reflect: What did you learn in this lesson? How do you know how to calculate the distance an accelerating object travels during some time interval? What do you need to know to be able to find that distance? PUM Kinematics Lesson 12: Position of an Accelerating Object as a Function of Time 77

78 13.1 Design an Experiment Lesson 13: Experimental Design Design an experiment to test whether the fan cart moves at constant speed or constant acceleration. Use available equipment. a) Describe the experimental setup in words and with a picture. b) List the quantities that you will measure and the quantities that you will calculate. c) Make measurable predictions for the outcome of your experiment based on the two different models of motion: constant speed (v = constant), constant acceleration (v = v 0 + at or v = at if v 0 = 0 ). How are the predictions different from the models? d) Perform the experiment, record the results, and decide which model you could not disprove and which model you were able to disprove Design an Experiment Design an experiment to determine the acceleration of the fan cart, first using sugar packets or any other mechanical things, and second using the motion detector. Compare the results. Start with the experiment that does not involve the motion detector. a) Describe the experimental setup in words and with a picture. b) List the quantities that you will measure and the quantities that you will calculate. c) Think about experimental uncertainties. Which instrument or which procedure will give you the largest uncertainty? d) Conduct the experiment and calculate the result. Express the value of the acceleration within a range. e) Then use the motion detector to measure the acceleration. f) Compare the results and account for the differences. Think about the assumptions that you made and the experimental uncertainties as a result of your equipment Communicating your results Homework Write a report for the experiments in 13.1 and Discuss what you learned about the acceleration of the fan cart. Use the rubrics below. 78 PUM Kinematics Lesson 13: Experimental Design

79 Scientific Ability Is able to distinguish between a hypothesis and a prediction. Is able to make a reasonable prediction based on a hypothesis. Is able to make a reasonable judgment about the hypothesis. Missing Hypothesis-prediction-testing rubric (for 13.1) No prediction is made. The experiment is not treated as a testing experiment. No attempt is made to make a prediction. No judgment is made about the hypothesis. An attempt A prediction is made but it is identical to the hypothesis. A prediction is made that is distinct from the hypothesis but is not based on it. A judgment is made but is not consistent with the outcome of the experiment. Needs some improvement A prediction is made and is distinct from the hypothesis but does not describe the outcome of the designed experiment. A prediction is made that follows from the hypothesis but does not have an if-andthen structure. A judgment is made and is consistent with the outcome of the experiment but assumptions are not taken into account. Acceptable A prediction is made, is distinct from the hypothesis, and describes the outcome of the designed experiment. A prediction is made that is based on the hypothesis and has an if-andthen structure. A reasonable judgment is made and assumptions are taken into account. Rubric for 13.2 Scientific ability Missing An attempt Needs some Is able to evaluate the results by means of an independent method. No attempt is made to evaluate the consistency of the results using an independent method. A second independent method is used to evaluate the results. However there is little or no discussion about the differences in the results due to the two methods. improvement A second independent method is used to evaluate the results. The results of the two methods are compared using experimental uncertainties. But there is little or no discussion of the possible reasons for the differences when the results are different. Acceptable A second independent method is used to evaluate the results and the evaluation is done with the experimental uncertainties. The discrepancy between the results of the two methods and possible reasons for the discrepancy are discussed. PUM Kinematics Lesson 13: Experimental Design 79

80 14.1 Observe and represent Lesson 14: Details of the Throw Throw an object up so that it comes close to the ceiling but it does not hit it. a) Describe the motion on the way up and on the way down. b) Draw a vertical motion diagram for the motion on the way up c) Draw a vertical motion diagram for the motion on the way down. d) Are the changes in velocity arrows consistent? e) Describe another real life situation that is similar to an object being thrown up into the air Observe and represent The data recorded in the table at the right are a record for the up and down motion of the center of a ball thrown upward (the y- axis points up). Fill in the table that follows. a) Draw a position-versus-clock reading graph b) Draw a velocity-versus-clock reading graph. Find its slope. What do you call this slope? c) How does the slope on the position versus time graph relate to the points plotted on the velocity versus time graph? d) Use the velocity versus clock reading graph to determine the ball s acceleration at the very top of its trajectory. Is the change in velocity consistent throughout the entire motion? On the way up? At the top? On the way down? e) What is the ball s velocity at the top? f) Can you reconcile the answers to parts (d) and (e)? Explain. Clock reading t (s) Position y (m) g) If the ball had zero acceleration and zero velocity at the top of its motion, what would happen to it? Did you know? The accepted value for the acceleration of objects close to Earth is 9.8 m/s 2, commonly referred to as "g". This was found experimentally in a manner close to the one you use in the problem above. 80 PUM Kinematics Lesson 14: Details of the Throw

81 h) How does your value for the acceleration compare to the accepted value? What might have caused any discrepancies? 14.3 Assumptions You are jumping off a high dive into a pool of water. It took you 1.5 sec to hit the water. a) Draw a motion diagram for your motion while you are in the air, ending several seconds after you enter the water. Are the motions similar or different? Explain. b) If you were to find the velocity with which you hit the water with, what assumptions do you make about the motion as you hit the water? c) If you dropped a water balloon from the football stadium grandstands which takes 1.8sec to hit the ground, what assumptions would you make about the final position and the objects motion? d) Discuss your answer with the class, are these valid assumptions, do they consider the motion of the object falling or the motion of the object sitting on the ground Reason and Represent A ball is hit by a baseball bat which is 1.2 m above the ground, straight up at 24.0 m/s. a) Sketch the situation to the top of the path, be sure to label and list all important givens and unknowns. State your assumptions. b) Draw a motion diagram and describe what happens to the speed as the ball approaches its maximum height. c) Determine all possible unknowns for the object when it reaches maximum height Regular Problem The fuel in a bottle rocket burns for 2.0 s. While burning, the rocket moves upward with an acceleration of 30 m/s 2. a) What is the vertical distance that the rocket travels while the fuel is still burning, and how fast is it traveling at the end of the burn? b) After the fuel stops burning, the rocket continues upward but is now slowing at a rate of about 10 m/s 2. Estimate the maximum height that the rocket reaches. What assumptions have you made in working through this problem? Homework 14.6 Regular Problem A ball is thrown upward at 20 m/s from the top of a 150 m building. PUM Kinematics Lesson 14: Details of the Throw 81

82 a) Determine all the information about the ball when it is at a height of 165 m. b) Determine all the information about the object when the ball just hits the ground Regular Problem James went outside and said he could throw a ball 25 meters upward. The ball takes 5.2 seconds to hit the ground. Show that James could throw at least 25 m up. Discuss any assumptions you made to determine this answer Reason Examine the graph below. a) Describe a real life situation that this graph could represent, be sure to include all the information on the graph and any extra in your situation. b) Determine the values of the physical quantities describing the motion during and at the end of the time interval. Discuss assumptions you made Reason You hold two identical golf balls. You drop one ball and simultaneously throw down the other ball. Explain what will be the same and different about their motion. a) Represent the motion of the balls with two motion diagram next to each other. b) Draw velocity versus time graphs for the two balls using the same axes. c) Draw position versus time graphs using the same axes. 82 PUM Kinematics Lesson 14: Details of the Throw

83 Lesson 15: Putting it all Together 15.1 Derive (very challenging!) So far we have been using two mathematical expressions when solving problems dealing with the motion at constant acceleration. v(t) = v o + a(t) x(t) = x o + v o (t) + ½a(t) 2 To find the displacement of an object, one must know the initial velocity, time interval and acceleration. However, we often do not know how long the motion lasted, but we have information about the final velocity (v f ) instead of the time. How can we use the initial velocity, the acceleration, and the final velocity of the object to find out how far it has traveled? Examine both equations above and think about how you can use the first one to derive the expression for the time interval, which you can then substitute into the second. Show your steps here. Hint 1: Write an expression for t using the first equation. Hint 2: Substitute the expression for t from the first equation into the second equation. This way you will eliminate t from the second equation. If you make no mistakes, you should get: v 2 = v o 2 + 2a(x-x o ) 15.2 Practice Use your new equation to solve the following problems: a) You throw a marble up at the speed of 10 m/s. How high will it reach? b) You drop a marble from a height of 1.5 m. What is the speed at which it will reach just before it hits the ground? c) A bus slows down from 15 m/s to 10 m/s in 200 meters. What is the acceleration of the bus? PUM Kinematics Lesson 15: Putting it all Together 83

84 d) A bullet traveling at a speed of 200m/s passed through a wooden block and gets stuck in it right as it is ready to come out. What is the acceleration of the bullet if the block is 40 cm wide? (pay attention to units here) 15.3 Represent and Reason Fill in the missing information on the graphs below. Match the description with the vertical set of graphs. a) An object decreasing speed then increasing speed in the opposite direction. b) An object increasing speed c) An object traveling at a constant velocity 15.4 Represent and Reason Fill in the missing information on the graphs below. For each set of vertical graphs, describe the motion for each segment. Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Time (sec) Time (sec) Time (sec) 84 PUM Kinematics Lesson 15: Putting it all Together

85 15.5 Summarize Analyze the information in the table below and complete the empty cells to summarize what you know about motion with constant velocity and with constant acceleration using the different representations of motion. Describe the Motion Motion at Constant Velocity Motion at Constant Acceleration. In words and provide an example. The object's velocity is increasing by the same amount every second. For example, a cart going down a smooth track that is tilted at an angle. With a motion diagram. With a graph of position versus clock reading. Mathematically as a function x(t). x(t) = x o + v o (t) + ½a(t) 2 PUM Kinematics Lesson 15: Putting it all Together 85

86 Describe the Motion Motion with Constant Velocity Motion with Constant Acceleration. With a graph of velocity versus clock reading. Mathematically as a function v(t). v(t) = constant With a graph of acceleration versus clock reading. Mathematically as a function a(t). a(t) = constant 15.6 Regular problem Homework A shuttle bus slows to a stop to avoid hitting a deer that had darted into the middle of the road. If the bus was initially traveling at m/s and had an average acceleration of m/s/s, what distance would it need to travel to avoid the collision? 86 PUM Kinematics Lesson 15: Putting it all Together

87 15.7 Represent and Reason Fill in the missing information on the graphs below. Then answer the questions that follow. I II III Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Position (m) Time (sec) Time (sec) Accel. Velocity (m/s) Time (sec) Time (sec) Time (sec) a) Which set(s) of the graphs represents the motion of an object thrown upwards? b) Which set (s) of graphs represents the object moving in the negative direction? c) Which set (s) of graphs represents the object first moving at constant speed and then slowing down? 15.8 Evaluate the Solution Identify any errors in the proposed solution to the following problem and provide a corrected solution if there are errors. The problem: A fire fighter slides down a fire pole at an increasing speed for 2.0 s, a distance of 2.0 m (she holds on so she doesn t move too fast at the bottom). She bends her knees at the bottom and stops in 0.10 m. a) Determine her speed at the end of the slide and just before she contacts the floor. PUM Kinematics Lesson 15: Putting it all Together 87

88 b) What is her acceleration while stopping? Proposed solution: v = v 0 + at = 0 + (9.8 m/s 2 )(2.0 s) = 19.6 m/s. a = (v 2 v 0 2 ) /2(x x 0 ) = [0 2 (19.6 m/s) 2 ]/2(0.10 m) = m/s Regular Problem A beach ball is volleyed up from a height 2.1m above the ground into up to a height of 13m. Find out the as many unknown physical variables about the initial conditions and 2.5 seconds later Regular Problem Heather and Komila are exercising in the park. When you start observing them, Komila is 50 m ahead of Heather. She is jogging at a speed of 5 mph and Heather is running at the speed of 7 mph. How long will take Heather to catch up with Komila? What assumptions did you make? Regular Problem This time Heather and Komila are running towards each other. How long will it take them to meet? What assumptions did you make? Reason Why is the head on collision of two cars more dangerous than the collision of cars traveling in the same direction? Pose a problem Pose a problem about a real life situation involving motion. Decide what information you can collect to solve it. Then use tools you learned in this module to solve it. Be prepared to present your problem to class Reflect Write a two-page summary of what you learned about motion so far. Make sure that you focus on the most important ideas. Give specific examples. The best summary will be used in the next edition of the module to help students learn kinematics next year. 88 PUM Kinematics Lesson 15: Putting it all Together

89 16.1 Regular Problem Lesson 16: Practice While concentrating on catching the football, a wide receiver on a football team runs into the goal post. He was originally moving at 10 m/s and bounced back at 2.0 m/s. A video of the collision indicates that it lasted s. Determine the acceleration of the receiver during the collision. Indicate any assumptions you made. How will you model the receiver to solve the problem? 16.2 Regular Problem While traveling in your car at 24 m/s, you find that traffic has stopped 30 m in front of you. Will you smash into the back of the car stopped in front of you? Your reaction time is 0.80 s and the magnitude of your car s acceleration is 8.0 m/s 2 after the brakes have been applied. List all assumptions you make Represent and Reason Assume that the positive direction of the x-axis is to the right. A car is moving according to the equation x = -30 (m) - 10 (m/s) t + 3(m/s 2 ) t 2 a) Describe the motion of the car in words. b) Determine the initial position, initial velocity, and acceleration of the car. Does it speed up or slow down? c) Draw a motion diagram for the car. d) Draw a velocity versus time graph for the car. Write a function v(t) for the graph. e) Sketch the position versus time graph right underneath the velocity versus time graph. What do you expect to see on this graph at the instant when the car stops? How far does it travel before it stops? f) How long does it take for the car to stop? What happens after that if the acceleration does not change? Is it a realistic situation? PUM Kinematics Lesson 16: Practice 89

90 16.4 Represent and Reason The graph below represents the position versus time graph for two rockets: A and B Rocket A Rocket B time See if anything massing on the graph. Add. Draw a picture of the motion of the two rockets. a) Calculate the slope of each line. What is the speed of each rocket? b) Write a mathematical expression x(t) for lines A and B. c) How much distance was traveled by rocket A after 6 minutes? d) How much time did it take rocket B to travel 60 km? e) How much sooner did rocket B travel 60 km than rocket A? f) If rocket A eventually landed at the same place it began, what would be the rocket s displacement for the entire trip? 16.5 Represent and Reason a) You ride your bike west at a speed of 8.0 m/s. Your friend, 400 m east of you, is riding her bike west at a speed of 12 m/s. (a) Fill in the table that follows. (Consider the bikes as point-like objects.) Sketch and translate: Draw a sketch of the initial situation and choose a coordinate system to describe the motion of both bikes. 90 PUM Kinematics Lesson 16: Practice

91 Represent physically: Draw a motion diagram for each bike. Represent mathematically: Construct equations that describe the positions of each bicycle as a function of time x(t). Solve and evaluate: Use the equations to determine when the bicycles are at the same position. Does your result make intuitive sense? b) After you fill in the table, draw on the left side below a position vs. time graph for each bicycle using the same set of axes. Position vs Time Graphs Velocity vs Time Graphs c) Are the slopes of the two lines and their initial values consistent with the actual motion and the coordinate system you used to describe the motion? d) Does the graph appear to correspond with the calculated answer for the time when the bicycles are at the same position? Explain. e) Beside the above position-versus-time graph, draw a velocity-versus-time graph representing the velocity of each bicycle on the same set of axes. f) Are the signs consistent with the word description of the motion? Explain. PUM Kinematics Lesson 16: Practice 91

92 92 PUM Kinematics Lesson 16: Practice

93 16.6 Real world application The following chart is taken from the NJ MVC driver s manual online m = 1.0 mi. 1.0 m = 3.28 ft a) What is the speed, in mph, for the car traveling in the problem above (16.2)? Was the answer close to the braking distance found in the NJ MVC driver s manual? If they are different, explain why. b) Explain why the reaction distances increase as the speed increases? c) Determine the reaction time they utilize in calculating the reaction distance in this table? Why might the NJ MVC choose this time? d) What is the car s acceleration (in m/s 2 ) for when it is traveling 70 mph? How does this compare to the acceleration in 16.2? 16.7 Reason a) Light from the Sun reaches Earth in about 8 minutes. If the speed of light is 3 x 10 8 m/s, how far is the Sun? Can we consider it a point like object in this problem? Can we consider Earth a point like object? b) The next closest star to Earth is Proxima Centauris. It is km away from the Solar system. If the star explodes tomorrow, when will see the explosion? 1 State of New Jersey Motor Vehicle Commission, (August 2008) PUM Kinematics Lesson 16: Practice 93