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. 12.2 Test Your Idea with Phet Simulations Go to http://phet.colorado.edu/en/simulation/moving-man and click on Run Now! We ve used the Moving Man before. Go back to Lesson 4 if you ve forgotten the details. 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. d) Predict the location of the moving man after 5 seconds. Show your work 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. 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 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. 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.
Velocity (m/s) Kreutter: Kinematics 12 12.3 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. 14 12 10 8 6 4 2 0-2 -4-6 0 20 40 60 80 100 120 Time (sec) Make sure you use the Problem Solving Strategy for the problems below. 12.4 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 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. 12.7 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. 12.8 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. Part I Part II +x Wants to stop: t o = 0 x o = 0 a 0,1 = 0 Foot gets to brake: t 1 = reaction time x 1 = position when foot hits brake a1,2 Finally stops: t 2 = time when car stops x 2 = final position when car stops 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? 12.9 Represent and Reason Homework 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. 12.10 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. 12.11 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 1 2 ( 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 d) Determine when and where the object will stop.
velocity (m/s) Kreutter: Kinematics 12 12.12 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. 12.13 Regular problem Examine the graph below. 25 20 15 10 5 0 0 1 2 3 4 5 6 7 Time (sec) 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? 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?