Lab 2 - Changing Motion

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Community College of Baltimore County *

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Physics

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Feb 20, 2024

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Save your work frequently! Name_______________________________________________Section______________________ LAB: CHANGING MOTION Learning goals:  Translate among the following representations of motion: verbal, motion diagram, and graphical: For example, given a verbal description of motion, create x - t , v - t , and a - t graphs; given a motion diagram, describe the motion in words, etc.  Given a v - t graph, determine average acceleration during a given time interval, and construct an a — t graph.  Determine change in velocity during a given time interval from an a — t graph.  Qualitatively and quantitatively reason with definition of acceleration.  Represent a physical quantity with a vector.  Add and subtract vectors graphically.  Use vector subtraction of velocity vectors and the definition of acceleration to determine the direction of acceleration. Materials:  Computer  Motion detector  Experiment files  Cart  Track  Fan unit In the previous lab, you looked at position-time and velocity-time graphs of the motion of your body and a cart at a constant velocity. The data for the graphs were collected using a motion detector. Your goal in this lab is to learn how to describe various kinds of motion in more detail. You have probably realized that a velocity-time graph is easier to use than a position-time graph when you want to know how fast and in what direction you are moving at each instant in time as you walk (even though you can calculate this information from a position-time graph). It is not enough when studying motion in physics to simply say that "the object is moving toward the right" or "it is standing still." When the velocity of an object is changing, it is also important to describe how it is changing. The rate of change of velocity with respect to time is known as the acceleration. To get a feeling for acceleration, it is helpful to create and learn to interpret velocity-time and acceleration-time graphs for some relatively simple motions of a cart on a smooth ramp or other level 1

Save your work frequently! surface. You will be observing the cart with the motion detector as it moves with its velocity changing at a constant rate. Investigation 1: Velocity and acceleration graphs In this investigation you will be asked to predict and observe the shapes of velocity-time and acceleration-time graphs of a cart moving along a smooth ramp or other level surface. You will focus on cart motions with a steadily increasing velocity. 1-1: Speeding Up In this activity you will look at velocity- time and acceleration- time graphs of the motion of a cart, and you will be able to see how these two representations of the motion are related to each other when the cart is speeding up. This could be done by moving the cart with your hand, but it is difficult to get a smoothly changing velocity in this way. Instead you will use a fan or propeller driven by an electric motor to accelerate the cart. 1. Set up the cart on the ramp, with the fan unit and motion detector as shown below. Be sure that the ramp is level. 2

Save your work frequently! 2. Open the experiment file called Speeding Up (L02Al-1) to display the axes that follow. 3. Use a position graph to make sure that the detector can "see" the cart all the way to the end of the ramp. You may need to tilt the detector up slightly. 4. Hold the cart with your hand on its side, begin graphing , switch the fan unit on LOW and when you hear the clicks of the motion detector, release the cart from rest. Do not put your hand between the cart and the detector. Be sure to stop the cart before it hits the end stop. Turn off the fan unit. Repeat, if necessary, until you get a nice set of graphs. Adjust the position and velocity axes if necessary so that the graphs fill the axes. Use the features of your software to transfer your data so that the graphs will remain persistently displayed on the screen . Also save your data for analysis in Investigation 2. (Name your file SPEEDUPl.XXX, where XXX are your initials.) 3

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Save your work frequently! 5. Insert your position and velocity graphs on the next page. Label the graphs "Speeding Up 1." (Ignore the acceleration axes for now.) 4

Save your work frequently! Question 1·1: How does your position graph differ from the position graphs for steady (constant velocity) motion that you observed in Lab 1: Introduction to Motion? Question 1-2: What feature of your velocity graph signifies that the motion was away from the motion detector? Velocity values that are positive, or have X > 0, signifies a positive increase in direction away from the detector Question 1-3: What feature of your velocity graph signifies that the cart was speeding up? How would a graph of motion with a constant velocity differ? There is an increasing or positive slope. A constant velocity graph would have a horizontal line with the y-value being equal to the velocity. Question 1-4: During the time that the cart is speeding up, is the acceleration positive or negative? How does speeding up while moving away from the detector result in this sign of acceleration? (Hint: Remember that acceleration is the rate of change of velocity. Look at how the velocity is changing. It takes two points on the velocity-time graph to calculate the rate of change of velocity.) Positive due to an increase in speed away/to the right/in a positive direction. Because of the direction and speeding up causing the velocity graph to have a positive slope, the acceleration sign would be positive. Question 1-5: How does the velocity vary in time as the cart speeds up? Does it increase at a steady (constant) rate or in some other way? Question 1-6: How does the acceleration vary in time as the cart speeds up? Is this what you expect based on the velocity graph? Explain. 5

Save your work frequently! Question 1-7: The diagram below shows the positions of a cart at equal time intervals as it speeds up. Assume that the cart is already moving at t 1 . At each indicated time, sketch a vector above the cart that might represent the velocity of the cart at that time while it is moving away from the motion detector and speeding up. Question 1-8: Show below how you would find the vector representing the change in velocity between the times 2 and 3 s in the diagram above. ( Hint: Remember that the change in velocity is what you need to add to the initial velocity to give the final velocity.) Based on the direction of this vector and the direction of the positive x axis, what is the sign of the acceleration? Does this agree with your answer to Question 1-4? Activity 1-2: Speeding Up More Prediction 1-1: Suppose that you accelerate the cart at a faster rate. How would your velocity and acceleration graphs be different? Draw your predictions in Logger Pro. 1. Test your predictions. Make velocity and acceleration graphs. This time accelerate the cart with the fan on HI. Repeat if necessary to get nice graphs. (Leave the original graphs persistently displayed on the screen.) When you get a nice set of graphs, save your data as SPEEDUP2.XXX for analysis in Investigation 2. 2. Insert your velocity and acceleration graphs below. 6

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Save your work frequently! Question 1-9: Did the shapes of your velocity and acceleration graphs agree with your predictions? How is the magnitude (size) of acceleration represented on a velocity-time graph? Question 1-10: How is the magnitude (size) of acceleration represented on an acceleration-time graph? Investigation 2: Measuring acceleration In this investigation you will examine the motion of a cart accelerated along a level surface by a battery driven fan more quantitatively. This analysis will be quantitative in the sense that your results will consist of numbers. You will determine the cart's acceleration from your velocity-time graph and compare it to the acceleration read from the acceleration-time graph. You will need motion software and the data files you saved from Investigation 1. 2-1: Velocity and Acceleration of a Cart That Is Speeding Up 1. The data for the cart accelerated along the ramp with half batteries and half dummy cells (Investigation 1, Activity 1-1) should still be persistently on the screen. (If not, load the data from the file SPEEDUPl.XXX.) Display velocity and acceleration, and adjust the axes if necessary. 2. Insert the velocity and acceleration graphs from Activity 1-1 below. Correct the scales if necessary. 7

Save your work frequently! 3. Find the average acceleration of the cart from your acceleration graph. Use the analysis feature in the software to read a number of values (say 10) of the acceleration, which are equally spaced in time. (Only use values from the portion of the graph after the cart was released and before the cart was stopped.) Acceleration Values (m/s 2 ) 1 0.204, 0.41 6 0.588, 0.63 2 0.233, 0.44 7 0.649, 0.40 3 0.376, 0.68 8 0.736, 0.59 4 0.423, 0.67 9 0.822, 0.47 5 0.515, 0.67 10 0.936, 0.49 Comment: Average acceleration during a particular time interval is defined as the average rate of change of velocity with respect to time-the change in velocity divided by the change in time. By definition, the rate of change of a quantity graphed with respect to time is also the slope of the curve. Thus, the (average) slope of an object's velocity-time graph is also the (average) acceleration of the object. 4. Calculate the slope of your velocity graph. Use the analysis feature of your software to read the velocity and time coordinates for two typical points on the velocity graph. (For a more accurate answer, use two points as far apart in time as possible but still during the time the cart was speeding up.) Velocity (m/s) Time (s) Point 1 0.05 m/s 0.208 s Point 2 0.78 m/s 1.564 s Calculate the change in velocity between points 1 and 2. Also calculate the corresponding change in time (time interval). Divide the change in velocity by the change in time. This is the average acceleration. Show your calculations below. Speeding up Change in velocity (m/s) Time interval (s) Average acceleration (m/s 2 ) Question 2-1: Is the acceleration positive or negative? Is this what you expected? 8

Save your work frequently! Question 2-2: Does the average acceleration you just calculated agree with the average acceleration you found from the acceleration graph? Do you expect them to agree? How would you account for any differences? 2-2: Speeding Up More 1. Load the data from your file SPEEDUP2.XXX (Investigation 1, Activity 1-2). Display velocity and acceleration. 2. Insert the velocity and acceleration graphs from Activity 1-2 on the next page. 9

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Save your work frequently! 3. Use the analysis feature of the software to read acceleration values, and find the average acceleration of the cart from your acceleration graph. Acceleration Values (m/s 2 ) 1 0.7s, 0.22 m/s 6 0.8, 1.48 2 0.73, 0.43 7 0.82, 1.65 3 0.76s, 0.64 8 0.84, 1.73 4 0.78, 0.85 m/s^2 9 0.85, 1.85 5 0.79, 1.24 10 0.91. 2.01 Average (mean) acceleration: m/s 2 4. Calculate the average acceleration from your velocity graph. Remember to use two points as far apart in time as possible, but still having typical values. Velocity (m/s) Time (s) Point 1 Point 2 Calculate the average acceleration. Speeding up Change in velocity (m/s) Time interval (s) Average acceleration (m/s 2 ) 10

Save your work frequently! Question 2-3: Does the average acceleration calculated from velocities and times agree with the average acceleration you found from the acceleration graph? How would you account for any differences? Question 2-4 : Compare this average acceleration to that with the fan cart on LOW (Activity 2-1). Which is larger? Is this what you expected? 2-3: Modeling a v-t graph Recall that in lecture we examined an x-t graph corresponding to walking at a constant rate (That is, changing position at a constant rate). We judged that the relation between x and t in this case is linear, and proceeded to find an equation for a straight line that serves as a mathematical model for the data. Now examine the v-t graph corresponding to a cart that’s changing velocity at a constant rate, from Activity 2-2. Is it apparent that this relationship is also linear? If so, we can model this data with an equation of the form y(x) = mx + b . 11

Save your work frequently! Question 2-5: Think about the independent and dependent variables. In math class, the independent variable is usually denoted by x , and the dependent variable by y . (The notation y(x) is sometimes used to remind us of this). But in this case, what physical quantities do the independent and dependent variables refer to? What symbols do we use for these quantities? Question 2-6: Determine the units of each of the variables and parameters (the slope and the y -intercept in a linear model). (Hint: Recall that every term in a polynomial must have the same units.) Question 2-7: How do we interpret the slope m in this case? (Confirm that the units are consistent with your answer.) Question 2-8: How about the intercept b ? (Confirm that the units are consistent with your answer.) 12

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Save your work frequently! Question 2-9: With your v-t graph from Act 2-2 selected, go to Analyze/Model, and click on “linear”. Model your v-t data, by adjusting the values of the parameters until the line passes as close as possible to the greatest number of the experimental data points. Write down the values of the parameters which give the best fit, and insert your graph below. 13

Save your work frequently! Question 2-10: Write down the equation that describes the particular motion you observed. Summarize your findings in the table below. Variables Parameters v t m b Units Physical interpretatio n Comment: You’ve now determined average acceleration 3 ways: (1) take values of acceleration at 10 random times and find the mean; (2) calculate using a =  v/  t for two data points; (3) modeling the data and determining the acceleration from the value of one of the paramenters. Investigation 3: Slowing down and speeding up In this investigation you will look at a cart moving along a ramp or other level surface and slowing down. A car being driven down a road and brought to rest when the brakes are applied is a good example of this type of motion. Later you will examine the motion of the cart toward the motion detector and speeding up. In both cases, we are interested in how velocity and acceleration change over time. That is, we are interested in the shapes of the velocity-time and acceleration-time graphs (and their relationship to each other), as well as the vectors representing velocity and acceleration. 3-1: Slowing Down In this activity you will look at the velocity and acceleration graphs of the cart moving away from the motion detector and slowing down. 1. The cart, ramp, and motion detector should be set up as in Investigation 1. Use the HIGH fan setting. The fan should be pushing the cart toward the motion detector. Now, when you give the cart a quick push away from the motion detector with the fan running, it will slow down after it is released. Prediction 3-1: If you give the cart a short push away from the motion detector and release it, will the acceleration be positive, negative, or zero (after it is released)? 14

Save your work frequently! Sketch your predictions for the velocity-time and acceleration-time graphs in Logger Pro. 2. Test your predictions. Open the experiment file called Slowing Down (L02A3-1) to display the velocity-time and acceleration-time axes that follow. 3. Begin graphing with the back of the cart near the 0.15-m mark. Tum the fan unit on, and when you begin to hear the clicks from the motion detector, give the cart a gentle push away from the detector so that it comes to a stop near the end of the ramp. (Be sure that your hand is not between the cart and the detector.) Stop the cart with your hand-do not let it return toward the motion detector- and turn the fan unit off immediately to save the batteries. You may have to try a few times to get a good run. Don't forget to change the axes if this will make your graphs easier to read. Move your data so that the graphs are persistently displayed on the screen . 4. Insert your predictions and graphs on the next page. 15

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Save your work frequently! Label your graphs with  A at the spot where you started pushing.  B at the spot where you stopped pushing.  C the region where only the force of the fan is acting on the cart  D at the spot where the cart came to rest (and you stopped it with your hand). Question 3-1: Did the shapes of your velocity and acceleration graphs agree with your predictions? How can you tell the sign of the acceleration from a velocity-time graph? Question 3-2: How can you tell the sign of the acceleration from an acceleration-time graph? Question 3-3: Is the sign of the acceleration (which indicates its direction) what you predicted? How does slowing down while moving away from the detector result in this sign of acceleration? (Hint: Remember that acceleration is the rate of change of velocity with respect to time. Look at how the velocity is changing.) Question 3-4: The diagram below shows the positions of the cart at equal time intervals. (This is like overlaying snapshots of the cart at equal time intervals.) At each indicated time, sketch a vector above the cart that might represent the velocity of the cart at that time while it is moving away from the motion detector and slowing down. Assume that the cart is moving at t 1 and t 4 . 16

Save your work frequently! Question 3-5: Show below how you would find the vector representing the change in velocity between the times 2 and 3 s in the diagram above. (Remember that the change in velocity is the final velocity minus the initial velocity.) Based on the direction of this vector and the direction of the positive x axis, what is the sign (the direction) of the acceleration? Does this agree with your answer to Question 3-3? Question 3-6: Based on your observations in this lab, state a general rule to predict the sign (the direction) of the acceleration if you know the sign of the velocity (i.e., the direction of motion) and whether the object is speeding up or slowing down. If you have more time, do the following Extension. Extension 3-2 Speeding Up Toward the Motion Detector Prediction 3-2: Suppose now that you start with the cart at the far end of the ramp, and let the fan push it toward the motion detector. As the cart moves toward the detector and speeds up, predict the direction of the acceleration. Will the sign (direction) of the acceleration be positive or negative? (Use your general rule from Question 3-6.) Draw your predictions for the velocity-time and acceleration-time graphs in Logger Pro. 1. Test your predictions. First clear any previous graphs. Graph velocity first. Graph the cart moving toward the detector and speeding up. Tum the fan unit on, and when you hear the clicks from the motion detector, release the cart from rest from the far end of the ramp. (Be sure that your hand is not between the cart and the detector.) Stop the cart when it reaches the 0.15-m line, and turn the fan unit off immediately. 2. Insert your predictions and graphs on the next page. 17

Save your work frequently! Question 3-7: How does your velocity graph show that the cart was moving toward the detector? Question 3-8: During the time that the cart was speeding up, is the acceleration positive or negative? Does this agree with your prediction? Explain how speeding up while moving toward the detector results in this sign of acceleration. (Hint: Look at how the velocity is changing.) Question 3-9: When an object is speeding up, what must be the direction of the acceleration relative to the direction of object's velocity? Are they in the same or different directions? Explain. Question 3-10: The diagram shows the positions of the cart at equal time intervals. At each indicated time, sketch a vector above the cart that might represent the velocity of the cart at that time while it is moving toward the motion detector and speeding up. Assume that the cart is already moving at t 1 . Question 3-11: Show below how you would find the vector representing the change in velocity between the times 2 and 3 s in the diagram above. Based on the direction of this vector and the direction of the positive x axis, what is the sign of the acceleration? Does this agree with your answer to Question 3-8? Question 3-12: Was your general rule in Question 3-6 correct? If not, modify it and restate it here. 18

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Save your work frequently! Question 3-13: There is one more possible combination of velocity and acceleration directions for the cart moving toward the detector and slowing down. Use your general rule to predict the direction and sign of the acceleration in this case. Explain why the acceleration should have this direction and this sign in terms of the sign of the velocity and how the velocity is changing. Question 3-14: The diagram shows the positions of the cart at equal time intervals for the motion described in Question 3-13. At each indicated time, sketch a vector above the cart that might represent the velocity of the cart at that time while it is moving toward the motion detector and slowing down. Assume that the cart is moving at t 1 and t 4 . Question 3-15: Show how you would find the vector representing the change in velocity between the times 2 and 3 s in the diagram above. Based on the direction of this vector and the direction of the positive x axis, what is the sign of the acceleration? Does this agree with your answer to Question 3-13? 19

Lab 2 - Changing Motion

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