# INVESTIGATION 1: VELOCITY AND ACCELERATION GRAPHS

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3 Lab 2 3 Question 1-4: 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 1-5: How does the acceleration vary in time as the cart speeds up? Is this what you expect based on the velocity graph? Explain. Question 1-6: During the time that the cart is speeding up, is the acceleration positive or negative? Explain how speeding up while moving in the positive direction results in this sign of acceleration (hint: remember that acceleration is the time rate of change of velocity)? Question 1-7: The diagram below shows the positions of a cart at equal time intervals. At each indicated time, sketch a vector above the cart that might represent the instantaneous velocity of each cart at that time. Then show in the space below the x-axis how you would find the vector representing the change in velocity between the times 1 s and 2 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 +x axis, what is the sign of the acceleration? Does this agree with your answer to Question 1-6?! Checkpoint 1 Activity 1-2: Speeding up more Prediction 1-1: Suppose that you accelerate the cart at a greater rate. How would your position, velocity and acceleration graphs be different? Sketch your predictions with dashed or different colored lines on the previous set of axes. Test your predictions. This time accelerate the cart with four batteries in the battery compartment instead of two. Leave your original graphs persistently displayed on the screen. When you get a nice set of graphs, save your data as speedup2 for analysis in Investigation 2.

4 Lab 2 4 Question 1-8: Did the shapes of your graphs agree with your predictions? If not, comment on the differences. Question 1-9: How is the magnitude (size) of acceleration represented on a velocity-time graph? How is the magnitude (size) of acceleration represented on an acceleration-time graph?! Checkpoint 2 INVESTIGATION 2: MEASURING ACCELERATION In this investigation you will quantitatively examine the motion of a cart accelerated along a level surface (that is, you ll get numerical results), using the files from the previous activity. Activity 2-1: Velocity and acceleration of a cart that s speeding up 1. Make sure the data from the first activity (the one with the data for the cart accelerated along the ramp with half batteries and half dummy cells (Investigation 1, Activity l-l) is still on the screen. (If not, load the data from the file SPEEDUPl.) 2. Find the average acceleration of the cart from your acceleration-time graph. Go under Analyze on the menu, then use Examine to select the portion of the graph you want to use. Then use Statistics to get an average and a standard deviation for your first activity (Run 1). (Only use values from the portion of the graph after the cart was released and before the cart was stopped.) Average (mean) acceleration (m/s 2 ) ± standard deviation (m/s 2 ) 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. 3. Click on the velocity-time graph to make it active. Go under Analyze and select Automatic Curve Fit; then select mt + b Linear to get a best-fit line to the appropriate portion of the Run 1 velocity-time graph. From the line s slope and the standard deviation in the slope, obtain the acceleration and its uncertainty. Average (mean) acceleration ± m/s 2 Question 2-1: Does the average acceleration you determined from the slope of the v vs. t graph agree with the average acceleration you found from the acceleration graph, within uncertainties? Is this what you expect? How would you account for any differences?

5 Lab Select the position-time graph. Go under Analyze and select Automatic Curve Fit; then select at 2 + bt +c Quadratic to get a best quadratic fit to the appropriate portion of the Run 1 position-time graph. Compare your result to the equation relating x and t for constant acceleration, x=v 0 t+1/2 a t 2 (note the a doesn t refer to the same thing in these two equations!). Use the appropriate coefficient from the best quadratic fit to determine the acceleration (the computer won t give you an uncertainty from this type of fit). Average acceleration ± m/s 2 Question 2-2: Does the average acceleration you determined from the fit to the x vs. t graph agree with the average acceleration you found from the acceleration graph, within uncertainties? Is this what you expect? How would you account for any differences? Activity 2-2: Speeding Up More Now repeat the above analysis for the latest data (where you used 4 batteries in the fan unit). From Statistics for the acceleration-time graph: Average (mean) acceleration (m/s 2 ) standard deviation (m/s 2 ) From a straight-line fit to your velocity-time graph: Average acceleration ± m/s 2 From a quadratic fit to your position-time graph: Average acceleration ± m/s 2 Question 2-3: Do the average accelerations obtained from the velocity graph and from the position graph 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 half batteries and half dummy cells (Activity 2-l). Is it significantly larger? Is this what you expected?! Checkpoint 3 INVESTIGATION 3: SLOWING DOWN AND SPEEDING UP In this investigation you will examine the motion of the cart toward the motion detector (in the x direction) and speeding up.

6 Lab 2 6 Activity 3-1: Speeding Up Toward the Motion Detector Prediction 3-1: Suppose that you start with the cart at the far end of the ramp, and let the fan push it toward the motion detector. Will the sign (direction) of the acceleration be positive, negative, or zero? Sketch your predictions for the velocity-time and acceleration-time graphs on the axes that follow. 1. Test your predictions. First clear any previous graphs. Click on the velocity graph to make it the active one. Graph the cart moving toward the detector (that is, in the negative x direction) and speeding up. Turn 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 0.5-m line, and turn the fan unit off immediately. Sketch these graphs in a different color or style on the axes above. Question 3-1: Which feature of your velocity graph shows that the cart was moving in the negative x direction? Question 3-2: 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 in the x direction results in this sign of acceleration. (Hint: Look at how the velocity is changing.)

7 Lab 2 7 Question 3-3: 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. Show below the x-asix how you would find the vector representing the change in velocity between the times 1 and 2 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 3-2? Question 3-4: When an object is speeding up, what must be the direction of the acceleration relative to the direction of the object s velocity? Are they in the same or different directions? Explain, using your results from this investigation and from investigation 1.! Checkpoint 4 Activity 3-2: Slowing down and speeding up In this activity you will look at what happens when the cart slows down, reverses its direction and then speeds up in the opposite direction. How does the velocity change with time? What is the cart s acceleration? The setup should be the same as for Activity 1-1. The fan unit should have the maximum number of batteries. Change the maximum time on your graphs to 10 s. Prediction 3-2: You start the fan and give the cart a push away from the motion detector. It moves away, slows down, reverses direction, and then moves back toward the detector. Try it first without using the motion detector! Now, for each part of the motion-away from the detector, at the turning point, and toward the detector-predict below whether the velocity is positive, zero, or negative. Also predict whether the acceleration is positive, zero, or negative. Moving away (+x) velocity: acceleration: At the turning point velocity: acceleration: Moving toward (-x) velocity: acceleration:

8 Lab 2 8 On the axes that follow sketch your predictions of the graphs of this entire motion. a (m/s^2) Time (s) 1. Test your predictions. Begin graphing with the back of the cart near the 0.5-m mark. Turn on the fan unit, 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 travels at least 1 m, slows down, and then reverses its direction and moves toward the detector. Be sure to stop the cart at least 0.5 m from the motion detector and turn off the fan unit immediately. You may have to try a few times to get a good round trip. Make sure the graph includes data showing what happened when the cart started and stopped, as well as when the cart was moving on its own. When you get a good round trip, sketch the graphs on the axes above using a different color or style. Question 3-5: Label the graphs with A where the cart started being pushed. B where the push ended (where your hand left the cart). C where the cart reached its turning point (and was about to reverse direction). D where you stopped the cart. Explain how you know where each of these points is. Question 3-6: What was the velocity of the cart at its turning point? Does this agree with your prediction? Question 3-7: According to your acceleration graph, what is the acceleration at the instant the cart reaches its turning point? Is it positive, negative, or zero? Is it significantly different from the acceleration during the rest of the motion? Does this agree with your prediction?

9 Lab 2 9 Question 3-8: Explain the observed sign of the acceleration at the turning point. (Hint: Remember that acceleration is the rate of change of velocity. When the cart is at its turning point, what will its velocity be in the next instant? Will it be positive or negative?) Question 3-9: How do your predictions compare to the graphs you got experimentally? Discuss any differences between them. Pay particular attention to the acceleration graph. Activity 3-3: Sign of Push and Stop On your acceleration graphs for the previous activity, look at the time intervals when you pushed the cart to start it moving and when you pushed to stop it. Question 3-10: What is the sign of the acceleration for each of these intervals? Explain why the acceleration has this sign in each case.! Checkpoint 5 INVESTIGATION 4: ACCELERATION OF FREELY FALLING OBJECTS Activity 4-1: Falling objects How does a freely falling object move? Investigate this using a motion sensor placed on the floor to track the motion of a falling object (place a wire cage over the motion sensor to protect it from the falling objects). Question 4-1: Does the object appear to move with constant acceleration? If so, find the value of the acceleration using the method from Investigation 2 that you think is best. Question 4-2: If you tape additional mass onto the object, will it have the same acceleration? Try it. Summarize the results. Question 4-3: Try dropping the object from different heights. Does this affect the motion? If so, how? Does it affect the value of the acceleration? Question 4-4: Try other types of objects. Do they all appear to fall with a constant acceleration? Is the value of the acceleration the same as for the original object? Do you see a pattern here?! Checkpoint 6

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