Lesson 3A: How Fast Are You Moving? 3.1 Observe and represent Decide on a starting point. You will need 2 cars (or other moving objects). For each car, you will mark its position at each second. Make sure the cars travel at two different rates. It wouldn t hurt to practice a bit first. 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 your slow car; record the data in the table. Repeat the experiment for the fast car and record that data as well. Clock Reading t Slow Car Position x Clock Reading t Fast Car Position x b) What are the physical quantities measured in this experiment? What units of measurement did you use? c) Explain the differences between these two ideas.
Distance (cm) Position (cm) Position (cm) Position (cm) Kreutter: Kinematics 3A 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 15 2 5 1 2 3 4 5 6 7 8 9 11 2 4 6 8 12 James Joe 15 2 5 2 4 6 8 12 2 4 6 8 12 b) 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 3.1. The position of the car is recorded on the vertical axis and the clock reading on the horizontal axis. c) Now draw a trend line for each car on the graph. What information can you learn about the motion of the car from the graph? Explain. 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? (Hint: look at the units!) c) Explain what it means if the slope is positive or negative. d) What is the velocity of each car in your
experiment? Explain how you know. e) How is velocity related to the concept of index you invented in lesson 2A? 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 Help? 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 some standard (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 Δ is the Greek letter delta and in physics and mathematics it is read as delta t (Δt) or the change in t. Time can be measured in many different units, such as seconds, minutes, hours, days, years, and centuries, etc. Clock reading: The clock reading or time (t) is the reading on a clock, stopwatch, or some other 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. 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. More help? Velocity of an object moving at constant velocity 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. Velocity is a vector quantity. Speed is the magnitude of velocity. Speed is always positive; it has no direction. Speed is a scalar quantity. 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. I m sure you 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 what s the independent and what s the dependent variable. For the problem below, the independent is t, and a dependent is x. Example: Examine: Define: Describe the relationship between the two variables. Describe the variables used in the scenario The object changes its position by 5 meters each second t = time elapsed x = position Time (second) Position (meters) 1 5 2 3 15 Represent: Write a mathematical equation using variables x = 5t This expression can be written as x(t) = 5t, however, in physics it is necessary to include units of measure x(t) = 5(m/s) t or x(t)=5(m/s)t. x(t) is read as x of t. 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.
Homework 3.4 Practice a) A car moved from x 1 2 mi to x 2 62 mi. Draw a picture with the coordinate axis, zero point and the locations x 1 b) A car moved from x 1 12 mi to x 2 34 mi. Draw a picture with the coordinate axis, zero point c) A car moved from x 1 3 mi to x 2 62 mi. Draw a picture with the coordinate axis, zero point d) A car moved from x 1 2 mi to x 2 78 mi. Draw a picture with the coordinate axis, zero point e) A car moved from x 1 62 mi to x 2 2 mi. Draw a picture with the coordinate axis, zero point 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 2 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) yards; III) 35 yards; IV) 25 yards. b) How far did the player travel from the beginning of observations? I) 2.5 yards; II) 2 yards; III) 35 yards; IV) 25 yards. c) What happened at the clock reading: I) The player started moving; II) The player was passing the mark of 45 yards;
Position (yard) Kreutter: Kinematics 3A 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. 6 5 4 3 2 5 15 2 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 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 = ( m/s) + t; II) x 1 = (5 m/s)t; x 2 = ( m/s)t; III) x 1 = t(5 m/s); x 2 = t( m/s); IV) both II and III are correct. c) How long will it take each object to travel 276 m? d) How far from each other will they be in seconds? 2 seconds after you start observing them? 3.8 Represent and Reason The motion of object A is represented by the function x A (5. 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 seconds? In minutes? In hours? 3. 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?.3 miles? 4 kilometers? What assumptions did you make? 3.12 Reason You walk 1.8 miles every 3 min. Use the index approach to calculate in your head how far you will walk in: (1) 1 hour; (b) 1 hour 3 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?