Question (12) Average velocity and position (update) for three hockey pucks You view the motion of a hockey puck in a video and mark its location every. The resulting image for three different hockey pucks is shown below. The hockey puck in each case is moving to the right. The gridlines in the images are 1 cm apart. Figure 1: A hockey puck travels to the right in three different cases. (a) In which case (A, B, or C) is there no interaction between the hockey puck and its surroundings? (Or rather, no net interaction.) (b) What is the (approximate) velocity of the puck in each case at t0? (Note: this is when the puck is at the location of the first image shown in the motion map. In each case, it is moving to the right.) (c) Why is your answer to the previous question an approximation and what could be done (in the experiment that produced the image) to make a better measurement of the velocity of the object at t0? (d) In Case (A), what do you predict will be the position of the puck at t2.0 s? (e) Is your prediction for the position of the puck at t2.0 s in Case (A) exactly correct or approximately correct? Explain. Solution (a) In which case (A, B, or C) is there no interaction between the hockey puck and its surroundings? (Or rather, no net interaction.) If there is no interaction, or no net interaction, then the momentum, and thus velocity, of the puck is constant. If the velocity of the puck is constant, then its displacement in equal time intervals is constant. The image in which the spacing between puck in equal time intervals is the same is Case (B). (b) What is the (approximate) velocity of the puck in each case at t0? (Note: this is when the puck is at the location of the first image shown in the motion map. In each case, it is moving to the right.)
Figure 2: There is no net interaction because the velocity of the puck is constant. Case (A) The best that we can measure is the average velocity of the puck from t0 to t and use this as an estimate of the initial velocity at t0. Use the gridlines to measure the position of the puck at t0 to t0.2 s as shown below. The gridlines are 1 cm apart. The time interval between successive images of the puck is. Figure 3: The instantaneous velocity at t0 is approximately equal to the average velocity from t0 to t. The average velocity of the puck from t0 to t is v avg r r f r i Since the puck is moving in a straight line in the x-direction, then the y-velocity and z-velocity is zero. Simply measure the x-velocity. 0.5 cm 0 2.5 cm/s The instantaneous velocity of the puck at t0 is approximately 2.5 cm/s. Since the puck is speeding up during its entire motion across the grid, then it s reasonable to guess that it is speeding up between t0 and t, though there s not enough data during that time interval to be certain. If it s speeding up between
t0 and t, then the instantaneous velocity of the puck at t0 is actually less than 2.5 cm/s. Case (B) Use the same procedure as for Case (A). However, note that the velocity of the puck is constant. Therefore, the instantaneous velocity at any instant is equal to the average velocity during any time interval. You can use any time interval to measure the average velocity of the puck and therefore know the instantaneous velocity of the puck at t0 exactly. Figure 4: Initial velocity at t0 is equal to the average velocity during any time interval since the puck s velocity is constant. 5 cm 0 25 cm/s Case (C) Use the same procedure as for Case (A). You will need to estimate the position of the puck at t. Figure 5: The instantaneous velocity at t0 is approximately equal to the average velocity from t0 to t.
9.3 cm 0 47 cm/s Again, this is the average velocity between t0 and t. The instantaneous velocity at t0 is approximately equal to the average velocity from t0 to t, which is 47 cm/s. However, when viewing the motion of the puck across the entire grid, it appears that the puck is slowing down. If it is also slowing down during the time interval from t0 to t, then the instantaneous velocity at t0 is actually greater than 47 cm/s. But with the data given, 47 cm/s is our best approximation. To make a better approximation, one needs to measure the position of the puck at smaller time intervals. (c) Why is your answer to the previous question an approximation and what could be done (in the experiment that produced the image) to make a better measurement of the velocity of the object at t0? To measure the instantaneous velocity at t0, we must measure the average velocity between t0 and t0.2 s and use this as an approximation for the instantaneous velocity at t0. To improve our approximation, we must measure the position of the puck at smaller time intervals. For example, if we measured the position of the puck every 0.05 s instead of every, then we would have a better approximation for the instantaneous velocity at t0. This reminds us that the instantaneous velocity equal to the average velocity in the limit as approaches zero. Thus, our approximation improves as gets smaller, i.e. approaches zero. (d) In Case (A), what do you predict will be the position of the puck at t2.0 s? The last image of the puck in Case (A) is at t1.8 s. The instantaneous velocity at t1.8 s is approximately equal to the average velocity between t1.6 s and t1.8 s. Figure 6: The instantaneous velocity at t1.8 s is approximately equal to the average velocity from t1.6 s to t1.8 s.
45 cm 35.5 47.5 cm/s Though the puck is speeding up, let s assume that the velocity is constant from t1.8 s to t2.0 s with the same velocity as from t1.6 s to t1.8 s. (This is our worst approximation, actually.) Then the velocity from t1.8 s to t2.0 s is approximately 47.5 cm/s. Use this as an estimate to calculate the position at t2.0 s. x f x i + v avg,x (45.0 cm) + (47.5 cm/s)() (45.0 cm) + (9.5 cm) 54.5 cm This is the equivalent of saying that if the puck travels 9.5 cm during the time interval from t1.8 s to t2.0 s then it will travel approximately 9.5 cm during the next time interval. But this assumes constant velocity, and clearly the puck is speeding up and it might be reasonable to assume that it will continue to speed up. As a result, our approximation is likely too small since in the next time interval, the puck s displacement will be more than 9.5 cm if it is speeding up. (e) As explained in the previous paragraph, our answer is an approximation. Indeed it s a poor approximation. To make a better approximation, you can make a graph of x-position vs. time. By fitting a best-fit curve to the data, you can determine x(t) and use it to calculate the x-position of the puck at t2.0 s.