22-April-2015: Collision in two dimensions

     The purpose of this experiment was to determine if momentum and energy are conserved in a two-dimensional collision.
     The setup consisted of two small, steel balls placed on a level table, one initially stationary and the other given some initial velocity. The ball with an initial velocity collides with the stationary one and they move away at some angle between them after the collision.
     The collision was recorded with a camera that was mounted above the table. Logger Pro's video analysis was used to capture and analyze the collision. Below are the camera settings used in Logger Pro.

     Data points were used to track each ball's position in the video. The origin was set to where the two balls collide, and also oriented so that the initial moving mass is traveling along the x-axis only. Subsequent position vs time graphs in the x and y-axis were produced.

Logger Pro's video analysis: Blue dots track the ball that had an initial velocity. Red dots track the ball that was initially stationary.

Below is the graph of the dots in the video analysis above. The blue and red dots are from the ball with initial velocity while the green and burgundy dots are from the stationary ball.

     To figure out if energy and momentum was conserved, equations were needed. The equations used were:

Note that there is no initial momentum or energy for the second ball because it is at rest.

     The balls had an x and y component when they moved so the velocities in each equation were substituted with the sum of the squares of the x and y velocities. The x and y velocities were found by analyzing the graph of the dots above. Only one example will be shown since all four trails of dots were analyzed the same way.
     Below is the position graph of the ball with an initial velocity. A linear fit was done before and after the zero position. This produced a slope for each linear fit which was the initial and final velocities in the x-direction for the first mass.

     After the velocities were found, simply plugging in the numbers and solving for each side of the equation was the next step. There were no units of length specified in this particular setup but the overall numbers should give the general idea of what was happening. The initial momentum and final momentum were calculated to be 12.40 units and 14.46 units, respectively. A difference of 2.06 units. Close, but this shows momentum was not completely conserved due to assumptions and errors. For now though, this data tells us that momentum was conserved in a larger frame of reference. Now, take a look at the energy data.

     The equation shown earlier for energy was used, and plugging in the numbers gave initial and final energies of 1150.54 units and 790.70 units, respectively. A difference of about 360 units. That is much more than the difference seen in the momentum part. Intuition tells us this difference is too big to consider energy to be conserved. 

     The data suggests that momentum can be conserved in a two-dimensional collision but that does not mean energy was conserved. Energy was lost during the collision to sound and heat. The glass table was assumed to be frictionless and therefore the calculations would not be perfect. The experiment was repeated again with a marble of different mass than the stationary steel ball. Although the results were better, the conclusion was the same. Momentum was conserved but energy was not.