Name: Date: Partners: LAB 2: ACCELERATED MOTION OBJECTIVES After completing this lab you should be able to: Describe motion of an object from a velocitytime graph Draw the velocitytime graph of an object from a verbal description of the motion Draw velocitytime and accelerationtime graphs given the positiontime graph Draw positiontime and accelerationtime graphs given the velocitytime graph Calculate acceleration from velocitytime graphs Give the direction of acceleration of an object by observing its motion OVERVIEW In the previous lab you examined and worked with positiontime and velocitytime graphs of your motion, or that of your lab partners, mostly when moving with constant velocity. In this lab we will concentrate on objects that will be speeding up or slowing down so it is not enough when describing the motion to simply say, the object is moving toward the right. Also, from the last lab you may have realized that although a velocitytime graph cannot give any information on the position of an object, it is better than a positiontime graph when you want to know how fast and in what direction you are moving at any instant in time. When the velocity of an object changes it is important to know how it is changing. This is best displayed on a velocitytime graph. The rate of change of velocity with respect to time is known as the acceleration. A graph will show at a glance if the velocity is changing steadily (constant acceleration) or if it is not changing steadily. In order to get a feel for acceleration, it is helpful to first create and learn to interpret velocitytime and accelerationtime graphs for some relatively simple motions. Since it is difficult to produce a steadily changing velocity when walking, you will use the motion of a fan cart on a track. This produces a motion in which the velocity increases or decreases steadily. Page 21
INVESTIGATION 1: VELOCITYTIME GRAPHS AND ACCELERATION In this investigation you will be asked to predict and observe the shape of the positiontime and velocitytime graph of a fan cart speeding up as it moves away from the motion detector. You will use the velocitytime graph to calculate the acceleration. Activity 11: Position, Velocity and Acceleration Graphs for Accelerated Motion. Procedure: 1. Set up the Fan Cart and Motion Sensor on the track as shown. Note that the fan can rotate in one direction, through 18 so that the air can be made to flow towards or away from the detector. Start with a simple motion. Observe the cart moving away from the detector and speeding up with the fan on high. Be sure to stop the cart before it hits the end stop and then turn off the fan to preserve the batteries. 2. Prediction: On the axes below use a dashed line to show the positiontime and velocitytime graphs you expect to get with the observed motion of the fan cart. Remember how the direction and sign of the velocity are related to the position of the sensor. Also recall your conclusion from lab 1: the velocity is the slope of the positiontime graph. Page 22
3. Open the file L2A11 to display positiontime and velocitytime axes with velocity from 1. to +1. m/sec and a time interval of 3. sec. If possible attach the motion sensor to the end of the track. You may need to tilt the detector up or down slightly to get the best results. Use a position graph to make sure that the detector can see the cart all the way to the end of the ramp. To do this you can look along the track and be able to see your reflection in the silvered surface of the motion detector. Also to check you can record data while moving the cart back and forth slightly at the far end of the track. The motion should be recorded faithfully on the graph. To preserve the batteries, only switch on the fan unit when you are making measurements and watch your fingers. 4. Have the cart speed up while moving away from the detector. Hold the cart about 5 cm from the sensor by placing your finger against the front end while the fan is on high. Make sure your hands and body are well clear of the sensor beam while recording. As soon as you release the cart double click the [START], button. Be sure to stop the cart before it hits the end stop and then turn off the fan to preserve the batteries. 5. Repeat until you get a nice set of position and velocity graphs. Draw the actual graphs you get on the same set of axes as your prediction using a solid line. Make sure you draw only that part of the curve that represents the motion of the cart as it speeds up from rest. Do not include the sudden change in velocity that occurs when you stop the cart. Question 11: How does the position graph differ from the position graphs for steady (constant velocity) motion observed in last week s lab? What feature signifies that the cart was speeding up? Question 12: What feature of your velocity graph signifies that the motion was in the positive x direction? Question 13: How does the velocity vary in time as the cart speeds up? That is, does it increase at a steady rate or in some other way? Explain your response. Page 23
6. Prediction: How would the graphs be different if the fan was on low instead of high? Draw your result with the fan on high on the axes below, and using a dashed line, your prediction for the fan on low. Briefly explain how the graphs will be different from the graph obtained with the fan on high. 7. Without deleting the results you got with the fan on high, use the same file L2A11 to record the positiontime and velocitytime graphs with the fan on low. Display both the results for high and low fan together so that they can be compared. Measuring Acceleration: Acceleration is defined as the change in velocity divided by the corresponding change in time, or simply the rate of change of velocity. Since the instantaneous accelerations cannot be measured directly, we have to use the velocitytime graph to calculate the acceleration. The slope of a velocitytime graph gives us the acceleration. A constant slope on a velocity time graph means the acceleration is constant. 1. Use the Smart Cursor,, to read the velocity and time coordinates for two typical points on your velocitytime graph. Remember you need to drag the cursor to the point you are measuring. (For an accurate result, you should use two points as far apart in time as possible but still during the time the cart was steadily speeding up.) Velocity Point 1 v 1 = t 1 = Point 2 v 2 = t 2 = Time Page 24
2. 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. Change in Δv = v 2 v 1 Time interval Δ t = t 2 t 1 Acceleration a = (v 2 v 1 )/(t 2 t 1 ) Just like the velocity, acceleration is a vector quantity and has magnitude and direction. It is a common error to assume that the direction of acceleration is the direction of motion. This concept causes much confusion so you will need to think about it carefully. The Direction of Acceleration: From the definition of acceleration, you can see that the direction of acceleration is the same as the direction of the change in velocity. To understand how we can get the direction of the change in velocity it is useful to represent velocities with an arrow pointing in the direction of motion it represents. The length of the arrow is drawn in proportional to the speed, so the greater the speed the longer the arrow. For example, in the accelerated motion activity you just did the positions of the fan cart at, 1, 2 and 3 seconds are given below. The velocities are increasing and can be represented by arrows drawn above the cart in each position. To distinguish velocity vectors we can use just one tick at the head of the arrow. Motion Detector Positive x direction The change in velocity between one and three seconds is given by: Δv = v 3 v 1 so v 3 = v 1 + Δv. This addition of vectors can be represented in a vector addition diagram in which arrows are drawn to scale to represent the direction and magnitude of the velocity vector. It is clear from our choice of +/ axes that the direction of the change in velocity is in the positive x direction, the same direction as the initial velocity. So Δv, the change in velocity, and the acceleration are positive, as is the slope of the velocitytime curve. So we find that when an object speeds up the velocity and acceleration are in the same direction. Page 25
Question 14: The acceleration is the slope of the velocitytime graph. Explain how both the velocity and acceleration graphs would show that the acceleration is constant. Question 15: Explain how both graphs would show the direction of the acceleration. Activity 12: The Direction of Acceleration for Other Kinds of Motion. In this activity you will determine the direction of the acceleration when the fan cart is slowing down as it moves away from the detector. You will also determine the direction of acceleration as it slows down or speeds up as it moves towards the detector. Procedure: 1. First observe the motion of the cart without recording data with the motion sensor. Rotate the fan so that the cart slows down as it moves away from the detector. You will have to push the cart to give it some initial velocity to the positive x direction. Catch the cart as it slows down to a stop at the end of the track so that it does not move back down the track. Motion Detector Positive x Direction 2. Draw velocity vector arrows above the carts at each position to show how the velocity changes. See the diagram in the previous activity to how it was done for an object speeding up. In this case assume the cart is already moving at t = s and has slowed to a stop at t = 3s. Indicate zero velocity with a dot. 3. Draw a vector addition diagram below to show how the velocity changes between s and 2s. To do this you have to draw the vectors v and Δv and show how they add to give the velocity vector, v 2. You can see how this was done in the previous activity except that in this case the change in velocity, Δv, between s and 2s is negative. Vector Diagram: Question 16: What is the direction of the acceleration for this motion? Explain. Here you see that if the direction of the acceleration and velocity are opposite the cart slows down. Page 26
4. Prediction: On the axes below use a dashed line to show the positiontime, velocitytime and the accelerationtime graphs you expect to get with the observed motion of the fan cart, slowing down while moving away from the detector. Don t forget that velocity is the slope of the positiontime graph, and the acceleration is the slope of the velocitytime graph. Discuss the graphs and try to come to an agreement as to what you should observe. 5. Open the file L2A12 to check your predictions. Draw the actual graphs with a solid line and draw only that part of the graph when the fan is pushing the cart. Don t change your prediction. If your prediction is incorrect you can avoid making the same error later if are able to reexamine the thought process that prompted you to make your prediction. Identify and label that part of the graph where the cart is pushed and then stopped. Question 17: a. What feature of the position graph shows that the cart is accelerating? b. What feature of the velocity graph shows that the cart is moving in the positive x direction? c. What feature of the velocity graph shows that the cart is slowing down? d. What feature of the acceleration graph shows that the accelerating is in the negative x direction? Page 27
Question 18: What do you conclude about the directions of the velocity and acceleration when the cart speeds up? Same/opposite? What do you conclude about their relative directions when the cart slows down? 6. Prediction: On the axes below use a dashed line to show the positiontime, velocitytime and the accelerationtime graphs you expect to get for a fan cart moving towards the motion detector when it is speeding up. Do the same for the cart moving towards the detector and slowing down. Don t forget that velocity is the slope of the positiontime graph, and the acceleration is the slope of the velocitytime graph. Discuss the graphs and try to agree on what you should observe. Cart Moving Towards Speeding up. Slowing down. Position (m) Position (m) Ti me ( s ) Ti me ( s ) Acceleration (m/s/s) Velocity (m/s) Ti me ( s ) Acceleration (m/s/s) Velocity (m/s) Ti me ( s ) Ti me ( s ) Ti me ( s ) 7. Use the file L2A12 to display position, velocity and acceleration graphs for these motions. Draw the actual graphs with a solid line on the same graph as your predictions. Don t change or erase your predictions. Page 28
Question 19: a. What feature of the velocity graph shows that both carts are moving in the x direction? b. What features of the velocity graphs show that one cart is speeding up and the other slowing down? c. What is the direction (+/ x) of the acceleration as the cart is moving in the x direction and speeding up? Explain. d. What is the direction (+/ x) of the acceleration as the cart is moving in the x direction and slowing down? Explain. Question 11: What do you conclude about the directions of the velocity and acceleration when the cart speeds up? Same/opposite? What do you conclude about their relative directions when the cart slows down? Accelerating While Slowing Down: You may have noticed that the term acceleration is used to describe the motion of an object that is speeding up or slowing down. The commonly used term deceleration is not used. Instead we find that for an object slowing down, the object is said to be accelerating, and that acceleration is in the opposite direction to the velocity. It is therefore the directions of the acceleration relative to the velocity that determines if the object speeds up or slows down. We can therefore state a fairly simple general rule. A useful general rule: When the acceleration is in the same direction as the motion (velocity), the object speeds up. When the acceleration is in the opposite direction to the motion (velocity), the object slows down. Question 111: Explain how the mathematical equation used to determine the velocity of an accelerating object, given below, is in agreement with this general rule. (i.e. What happens to the absolute value of v if v and a are in the same direction and therefore both positive or both negative? What happens to the absolute value of v if v and a are opposite in direction and in sign?) Equation: v = v + aδt Page 29
Vector Representation of Acceleration. In any diagram of a moving object it is useful to indicate the direction of the velocity and acceleration. You have already used an arrow with a single tick to represent the direction of the motion or velocity. Using the idea expressed in the above paragraph, an arrow giving the direction of the acceleration would also tell whether the moving object is speeding up or slowing down. To distinguish this from the arrow representing the velocity a double tick can be used as shown above the cart in the figure below. Question 112: Use the general rule given above to determine if the cart pictured on the right is speeding up or slowing down. Explain your choice. Activity 13: Position, Velocity and Acceleration for a Cart that Turns Around. 1. Prediction: On the axes below use a dashed line to show the positiontime, velocitytime and the accelerationtime graphs you expect to get for a fan cart moving away from the motion detector if the fan causes it to slows down, turns around and speeds up moving back towards the detector. Again recall that that velocity is the slope of the positiontime graph, and the acceleration is the slope of the velocitytime graph. Observe the motion without recording data. Note: you will have to give the cart a push so that it has an initial velocity away from the motion detector with the fan pushing in the opposite direction. Thereafter the motion is determined by the fan alone, until you stop it so that it does not crash into the detector. Discuss the graphs and try to agree on the shapes of the graphs you should get. Moving Away and Turning Around Page 21
2. Use the file L2A12 to display position, velocity and acceleration graphs for this motion. In these graphs record the push and the stop. Draw the actual graphs with a solid line on the same graph as your predictions. Don t change or erase your predictions. Question 113: Does the velocity change direction and sign in these motions? Does the magnitude of the velocity change? What is the velocity at the turning point? Question 114: Does the acceleration change direction and sign in these motions? Does the magnitude of the acceleration change? What is the acceleration at the turning point? 3. Draw the velocity and acceleration vectors above the carts for this motion as shown on page 29. At position 1 the cart is moving away and slowing down. At position 2 it is at the turning point and in position 3 it is returning and speeding up. Make sure the direction and magnitude of the vectors agree with your graphs and your answers to the questions above. Moving Away and Turning Around Question 115: Does the direction and sign of the velocity and acceleration vectors agree with the general rule for the relative direction of those vectors when an object speeds up and slows down. That is, are those vectors in the same direction when the cart speeds up and opposite when it slows down. Page 211
Conclusion: (Approximately 1 page in length. Use separate sheet of paper.) Questions: 1. For each of the positiontime graphs given below, sketch below its corresponding velocitytime and accelerationtime graphs. Describe the motion you have graphed above. Use the terms: moving to the right/left (or in the positive/negative x direction) and moving steadily or speeding up/slowing down steadily. Motion 1: Motion 2: Page 212
2. A car can move along a straight line on the + or xaxis. Sketch the velocitytime graph that corresponds to each of the following descriptions of the car's motion. Assume that the positive x direction is chosen to be to your right and do not attempt to represent the instantaneous changes of velocity on the acceleration graph since it an infinite acceleration. A car moves to your right for 6 seconds at constant speed and then is stationary for 4 seconds. A car moves to your left for 6 seconds at constant speed and then slows steadily to a stop in the next 4 seconds. A car moving to your right slows down steadily to a stop until its velocity is zero after 8 seconds. A car moving to your right slows down steadily for 6 seconds until its velocity is zero, at which point it turns around and starts back speeding up steadily Page 213
3. a. Draw the velocitytime and acceleration time graphs for a ball that is thrown vertically into the air and falls back down to its initial height. Assume that the +x axis is up. On both graphs show the instant at which the ball reaches its highest point. b. Based on your understanding of the motion of falling objects, what is the numerical value of the slope of the velocitytime graph for an object thrown in the air? Explain. c. What is the slope of the velocity graph at the highest point? What is the acceleration at that instant? Give a numerical value. Page 214