The purpose of this experiment was to prove that the conservation of momentum theory is true and applies to mostly elastic two-dimensional collisions. The purpose was also to see if energy is conserved in such a collision as well.
Apparatus:
Figure 1: Apparatus of this experiment
The apparatus of this experiment consisted of a large horizontal tabletop that has a glass surface. The collision occurred on this tabletop since it is level, ensuring that there is no net external force on the system for conservation of momentum and energy to occur. In addition, it was used in order to make friction negligible between the balls and the surface. A tall steel vertical rod with a horizontal rod attached to it was used to allow the camera to record from birds eye view to ensure correct measurements of the displacement of the ball before and after the collision. The camera, attached to the horizontal rod, took a recording of the collision of the metal balls and their displacement. A computer with Logger Pro software saved the recording as a movie capture for data collection and analysis. Two steel balls of same mass and an aluminum ball were used for the collision, shown below:
Figure 2: Two steel balls (to the left) and aluminum ball (to the right) used in this experiment
Lastly, the masses of the balls were obtained using the accurate balance shown below:
Figure 3: Balance used to measure the masses of the metal balls
Procedure:
Once all of the equipment was obtained, the camera was attached to the computer containing the Logger Pro software and the camera was run through Logger Pro's "Movie Capture" application. Then, the masses of the two steel balls and the aluminum ball were obtained via the balance shown in figure 3. The mass of the colliding steel ball was measured to be 66.7 grams, and the mass of the stationary steel ball was found to be 66.8 g. Lastly, the mass of the colliding aluminum ball was recorded as 28.9 grams.
Next, two collisions of the metal balls were captured as videos. The first collision occurred between the colliding steel ball and stationary steel ball. The colliding steel ball was the ball that contained the initial momentum and kinetic energy and collided with the other steel ball. The stationary steel ball was the ball that was initially at rest and gained momentum after the collision with the colliding steel ball. The stationary steel ball was set at about the middle of the glass tabletop and it was ensured that it would not roll by itself by leveling the tabletop to be horizontal. Then, the colliding steel ball was rolled from one of the sides of the glass tabletop and was directed such that it collided with the stationary steel ball.The goal was to make the balls roll off at decent angles from one another after the collision. The same was performed for the second collision, with the only difference that the colliding steel ball was replaced with the colliding aluminum ball.
Once the video captures were recorded, the video analysis application in Logger Pro was used to obtain the x and y displacements as well as the x and y velocities of the colliding balls before and after the collisions as well as the x and y displacements/velocities of the stationary balls after the collisions. The x and y measurements were obtained based off of the placing of the origin and the rotation of the axes. The origin was placed at the place of collision (i.e. where the stationary ball was before and during the collision) and the axes were rotated such that one of the axes was corresponding with the direction of motion of the colliding ball before the collision. The video capture screenshots of both trials are shown below:
Figure 4: Video analysis of the collision of Trial 1 between the two steel balls
Figure 5: Video analysis of the collision of Trial 2 between the aluminum and steel balls
Using the x and y displacement/velocity data, the momentum of both individual metal balls in the x and y directions as well as the sum of the momentum of both balls in the x and y directions were found. The total momentum and the kinetic energy of the whole system was also found using the x and y velocity data fore both trials.
Data Analysis:
The momentum of one ball in the x direction was found by multiplying the mass of the ball by the x velocity obtained:
The previous calculations found for both trials are displayed in the following tables:
Table 1: Momentum and KE calculations of trial 1
Table 2: Momentum and KE calculations of trial 2
Graphs of the momentum and kinetic energy of the system of trial 1 before, during and after the collision are displayed below.
Graph 1: Momentum in x and y direction and total momentum of the system of trial 1
Graph 2: Kinetic Energy of the system of trial 1
Graphs of the momentum and kinetic energy of the system of trial 2 before, during and after the collision are displayed below.
Graph 3: Momentum in x and y direction and total momentum of the system of trial 2
Graph 4: Kinetic Energy of the system of trial 2
Discussion:
Looking at graph 1, the momentum in the x direction of the system of trial 1 has a mean of -22.02 N*s with a standard deviation of 0.68. This momentum makes sense to be at such a value since momentum in the x direction was initially large, and, by the conservation of momentum principle, it will be constant during and after the collision. In addition, the standard deviation is small (around 3.0%), proving that the conservation of momentum existed in the x direction. Next, the momentum in the y direction of trial 1 has a mean of 2.272 N*s with a standard deviation of 0.563. The value of the momentum in the y direction makes sense because the momentum in the y direction is so small, corresponding to the orientation of the axes during the video analysis. That is, if the x axis was proportionally corresponding to the direction of the motion of the colliding ball, the momentum in the y direction would be zero, assuming ideal conditions and no sources of error/uncertainty. Since the x axis was not completely proportional to the motion of the colliding ball and since the conditions aren't ideal and there exists error and uncertainty, the momentum in the y direction is not zero. On the other hand, the standard deviation is fairly large (about 24.8%) for the momentum in the y direction. However, that large standard deviation is caused solely by the uncertainties and errors in this experiment (explained in the conclusion). Since the value of momentum in the y direction is small, the uncertainties and errors of this experiment have a larger effect on its accuracy and precision, resulting in a larger deviation. This is seen by noticing that the standard deviation of the momentum in the x and y direction are almost equal. If all of the uncertainty and error would be made smaller, the standard deviation of the momentum in the y direction would be smaller. Taking into account the effects of the uncertainty and error on the value of the momentum in the y direction and ignoring them, it is shown that the momentum in the y direction was also conserved. Since the momentum in the x and y direction were conserved, it can be seen that the total momentum of the system of trial 1 is also conserved. In addition, the expected mean of the total momentum (22.14 N*s), depending on the x and y momentum values, as well as the small and nearly equal standard deviation (0.67) also prove that the momentum of the whole system in trial 1 was conserved.
Looking at graph 2, the mean of the kinetic energy of the system in trial 1 was 3066 J, and the standard deviation was 154. Since the standard deviation is small (5.0%), it can be seen that kinetic energy was also conserved in this system. However, the value of the kinetic energy is unexpectedly large, especially for the system used in this experiment). The large kinetic energy resulted from the large values of velocity in both directions for both metal balls obtained from the video analysis. It is unclear why the velocities turned out to have such large values, but it is most likely caused by an error in the movie capture and video analysis application. However, the purpose of the experiment was achieved by proving that the kinetic energy of the system is conserved.
Looking at graph 3, the momentum in the y direction of the system of trial 2 has a mean of 7.491 N*s with a standard deviation of 0.764. This momentum makes sense to be at such a value since momentum in the y direction was initially large, and, by the conservation of momentum principle, it will be constant during and after the collision. The value of the momentum in the y direction makes sense because the momentum in the y direction is large and the momentum in the x direction is small, corresponding to the orientation of the axes during the video analysis. In this analysis, the y axis was the axis that corresponded to the velocity of the colliding ball before the collision. In addition, the standard deviation is fairly small (around 10.2%), proving that the conservation of momentum existed in the y direction. The reason the standard deviation is larger is because the uncertainty and error in the system of trial 2 is larger, due to the fact that a ball of smaller mass and composed of different and less dense metal was collided with a larger and denser metal ball, resulting in the collection of smaller values and making more room for error. In addition, smaller data leads to smaller results, which are more sensitive to any error than larger results. Next, the momentum in the x direction of trial 1 has a mean of 1.604 N*s with a standard deviation of 0.439. For the same reason for the momentum in the y direction being large, the momentum in the x direction is small because of the fact that the y axis was corresponding to the motion of the colliding ball before the collision. In addition, the momentum in the x direction is not zero because the y axis was not directly proportional to the motion of the colliding ball, the conditions aren't ideal and there exists error and uncertainty. On the other hand, the standard deviation is fairly large (about 27.4%) for the momentum in the x direction. However, as stated for graph 1, since the value of momentum in the x direction is small, the uncertainties and errors of this experiment have a larger effect, resulting in a larger deviation. Ignoring the effects of the uncertainty and error on the value of the momentum in the x direction, it was also conserved. Since the momentum in the x and y direction were conserved, it can be seen that the total momentum of the system of trial 2 is also conserved. In addition, the expected mean of the total momentum (8.116 N*s) as well as the fairly small standard deviation (0.734) also prove that the momentum of the whole system in trial 2 was conserved.
Lastly, analyzing graph 4, the mean of the kinetic energy of the system in trial 2 was 584.5 J, and the standard deviation was 95.1. Since the standard deviation is proportional to the error and uncertainty of trial 2 , although a little larger (16.3%), it can be seen that kinetic energy was also conserved in this system. The reason that the standard deviation is a little larger is because the uncertainties and errors in the values for the two metal balls were squared , halved, and added. Just like in graph 2, the KE values are also unexpectedly large, caused by the large velocity values. It is also unsure why the KE is also large in this case, but it also has to do with an error in the video analysis application in Logger Pro. However, it was seen that the conservation of energy principle applies to this system.
Conclusion:
The purpose of this experiment was to prove that the conservation of momentum and conservation of energy applies to elastic collision. This was achieved by showing that, ignoring the effects of uncertainty and error, the kinetic energy and momentum values of the elastic collision in this experiment were constant throughout the experiment, proving that the kinetic energy and momentum were conserved.
Some sources of error and uncertainty that have a large effect on the results in this experiment include:
- the measurement of displacement and velocity of the metal balls throughout the experiment. Due to the low quality of the camera as well as the small size of the metal balls, it was difficult to exactly pinpoint the center of the balls and be consistent with it. In addition, it is unknown why the velocity values and the displacement values came out to be very large, not giving accurate data.
- The camera was far from the tabletop, making it harder to also pinpoint the center of balls. In addition, from that height, the difference in displacement was very small between each frame, making it difficult to add points that were consistent with the motion of the metal balls, especially after the collision.
- The spinning and sliding of the metal balls during the experiment. Sliding and spinning would have an effect before the collision, and spinning would have an effect after the collision.
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