22-Apr-2015: Collisions in two dimensions

PURPOSE

The purpose of this experiment was to analyze two dimensional collisions to see if momentum and kinetic energy are conserved.

PROCEDURES

Figure 1: Set-up of experiment

Figure 2: Balls used to simulate collisions

This experiment was conducted with the set-up displayed in Figure 1. The camera (circled in red) was used to capture the motion of two metal ball before and after a collision on a glass surface (circled in blue). As shown in Figure 2, two of the balls that we used in this experiment were the same size and mass, while the third was smaller and lighter. In the first part of the experiment, we used the two balls that were equal in mass. We placed one of them near the center of the surface and rolled the other towards it to cause a collision.

Figure 3: Graphs of position vs time

After capturing the motion of the balls with the camera, we used video analysis on Logger Pro to
track their positions in the x- and y-directions throughout the entire process. Then, we plotted their positions with respect to time. The location marked by the black circle indicates when the collision occurred. Therefore, any graphs before the black circle refer to the initial velocities, while the the graphs after the black circle correspond to the final velocities. The red graph in Figure 3 correspond to the velocity of the rolled ball in the x-direction, while the blue graph correspond to its velocity in the y-direction. The green and brown graphs correspond to the stationary ball's velocity in the x- and y-directions, respectively (notice that the green and brown graphs begin after the collision since the ball was at rest until that event).

Figure 4: Calculations of initial and final momentum

Figure 5: Calculations of initial and final kinetic energy

After finding the velocities, we calculated the momentum of the system before and after the collision in both the x- and y-directions (Figure 4). As it can be seen from the figure, the initial momentum of the stationary ball was zero in both cases since it did not have any velocity. When we completed the calculations, we compared the values on both sides to see if they were equal. The percent error in the x-direction was only 5.97 percent, while the percent error in the y-direction was 35.97 percent. Although the percent error was technically extremely high in the y-direction, we can assume that momentum was mostly conserved since the momentum values in the y-direction were much smaller than the values in the x-direction. The reasons why these values were not a hundred percent accurate will be discussed later in the conclusion.

In addition to seeing if the momentum was conserved, we also saw if the kinetic energy was conserved (Figure 5). In this case, we did not have to break the kinetic energy values into components because energies are scalars. This time the percent error was much higher at 25.25 percent. However, this is to be expected because some energy was most likely lost in the form of heat and sound when the balls collided.

Figure 6

For the second part of the lab, we rolled the less massive ball at a larger, stationary ball. We repeated the process from the first part of the lab and used video analysis to track the motion of the balls. We once again graphed their positions in the xy-plane with respect to time and applied "Linear Fit" on each plot of points to find the velocities. This graph is shown above in Figure 6. The moment at which the balls collided is marked with the black circle.

Figure 7

Figure 8

Using the initial and final velocities we found by analyzing the graph in Figure 6, we were able to find the initial and final momentum of the system in the x- and y-directions. We were also able to find the initial and final kinetic energy. The mathematical process is shown above in Figures 7 and 8. This time the percent error in the conservation of momentum equation in the x-direction was a bit higher at 12.28 percent. This may be due to the friction between the stationary ball and the surface, which was probably more significant than Part 1 of the experiment because the smaller ball most likely did not exert as much force on the stationary ball as the bigger ball of Part 1 did. In addition, the percent error in the y-direction was 43.72 percent. As mentioned before, this error can be ignored because the momentum in the y-direction was very small compared to the momentum in the x-direction. Moreover, the percent error for the kinetic energy values was 38.49 percent. Again, this is to be expected because some of the initial kinetic energy probably dissipated during the collision in the form of heat and other forms of non-conservative work.

CONCLUSION

From this experiment, we learned firsthand that momentum is conserved in two dimensional collisions, while kinetic energy is not. This is because some of the energy is lost in the form of heat and sound. In addition, we gained valuable experience in using video analysis to track the motion of objects in two dimensions. This will be useful skill that we can employ in the future.

Some error could have come from the video analysis since it was difficult to accurately track the motion of the balls. Therefore, the velocity values were most likely a bit off. Moreover, we did account for the fact that there were some friction between the balls and the surface that they were rolling on. This could have also contributed to the error.