Putting Around with Momentum

By: Shaila Murthy

For my example of conservation of momentum, I initially wanted to roll a snowball at another object to create an inelastic collision, however I was not able to create one that would roll well enough to actually create the scenario. Instead, I decided to use two golf balls because I knew they would both be able to roll nicely. I placed them on my driveway, which has a slight incline, and ended up using my foot to nudge the first ball along (I was home alone when I did this).

After I had a successful run (following many failures in which the first golf ball just plain missed the second one), I inserted the video into LoggerPro to analyze it and find the velocities. I found the velocity of the first ball from the beginning of its movement to the point at which the second ball left the video, and the second ball from right before the collision to the point at which it left the video as well.

Here is the (mildly excessive) table of values. The red and blue values are the first ball, and the green and maroon are the second. This does not show the beginning of the first ball's movement, simply because it did not end up being relevant for the calculations.

Here is the graph of the x-velocities of the two balls (once again, red is the first ball and green is the second). The collision is actually quite clear, as it occurred right where the red and green triangles are at almost the same velocity (at roughly 2.2 seconds).

In the graph and the table, it appears as though the second ball was moving prior to the collision, however that was simply due to the position of the ball vs. the camera moving to catch the motion. After I had found the velocities of the balls, I began my calculations. I started with finding the angle at which the balls were inclined. I measured the distance between the two balls to be 61 cm, and the horizontal distance to be 60 cm (which i found by holding a tape measure horizontally across the distance rather than pressed against the pavement). From there I used right triangle math to calculate my angle to be 10.4^o. 

Once I found this, I began to plug values into my momentum equation. The mass of both the balls was equal to 0.046 kg, and I used the initial velocity for ball one of 0.226 m/s, the final velocity of ball one of 0.142 m/s, and the final velocity of ball two of 0.127 m/s.

m₁v₁ = m₁v_1f + m₂v_2f 

(0.046)(0.226)(cos(10.4)) = (0.046)(0.142)(cos(10.4)) + (0.046)(0.127)(cos(10.4))

Here is where I realized that having the balls on an incline actually did not add any new element into my calculations. I had initially thought that it may make the process more interesting, however I realized that the balls continued down the same incline after their collision, so the cosines simply cancelled out (meaning my lovely triangle math was all irrelevant in the end). By cancelling out the cosines and masses, I ended up with this:

(0.046)(0.226)(cos(10.4)) = (0.046)(0.142)(cos(10.4)) + (0.046)(0.127)(cos(10.4))

0.226 = 0.142 + 0.127

0.226 /= 0.269

Obviously the collision was not perfectly elastic, and was rather actually almost over-elastic (which is absolutely a term I just made up, since inelastic means losing kinetic energy and this collision gained kinetic energy). I attribute this anomaly to the fact that they were on an incline, the likeliness of friction playing a role, and also to the fact that my point-marking skills on LoggerPro are far from perfect. So, unfortunately, I did not really show conservation of momentum. However, in a perfect world where I had another person to help me with this so I didn't have to use my foot and an inclined slope, where there was no friction, and where I was a "pro" at LoggerPro, this should have showcased conservation of momentum. 

Even though my calculations found that momentum was gained, kinetic energy was still lost during the collision, and I was able to calculate the amount and the percentage that was lost.

1/2mv₁² = 1/2mv_1f² + 1/2mv_2f² 

(0.226)² = (0.142)² + (0.127)²

0.0511 = 0.0202 + 0.0161

0.0511 /= 0.0363

0.0148 J Lost

(0.0148/0.0511) * 100 = 28.96% KE Lost

Overall, from this assignment I unfortunately did not accurately prove conservation of momentum, however I was still able to calculate the amount of kinetic energy lost. In a perfect world, this would have actually shown conservation of momentum, however the world is not perfect and neither are any of us, so here we are.