24-April-2017: Collisions in Two DImensions

Lab 15: Magnetic Potential Energy Lab

24-April-2017
Idea: Create two-dimensional collisions and verify whether momentum and energy are conserved.

Summary:

1) Using a leveled glass table, I set up my phone camera that will capture the collision of

1. Steel ball with steel ball, one is initially stationary
2. Steel ball with smaller aluminum ball, also initially stationary, at 240 fps.

2) Thus the setup looked like such:

3) I record two videos for the two cases, and export the videos to Logger Pro, where I begin to plot the position of the two balls. Then, I scale the image based on the width of the glass table, and position an xy-axis over the video in order to create my graphs, like the following.

4) Therefore this gives me four sets of data for each of the two test cases: positionof x before and after the collision for both balls, position of y before and after the collision for both balls. Because I am trying to verify whether momentum & energy are conserved, what is important here is only the components of velocity in the x-y direction before & after the collisions: these will be used to calculate the momentum and energy, and as such the slopes of the position-time graphs gives me these velocities.

5) Lastly, I produce graphs of the position x-y center mass for both systems throughout the collisions, and also the velocity x-y center mass for both systems.

Data Analysis/Discussion

1)  In order to demonstrate that momentum is conserved, I obtain the slopes of the position-time graphs, which would allow me to find my values of velocity for the balls. After collecting this data, I calculated for momentum by writing out momentum (M=mv) before and after the collision, and the results I got did indeed verify this, getting results for Momentum initial versus Momentum final that were relatively close to each other:

Steel vs Steel ball

Mx: 0 = -.0508 N*S

My: .6548 = .5367 N*S

Steel vs Aluminum Ball:

Mx: .00556 = .0059 N*S

My: -.0163 = -.01459 N*S

To verify this further, I then performed percent error calculations, which confirmed this (12-27%), exception being for the momentum in the x direction for the balls of equal mass--since the way I positioned my axis meant that the initial x-component of velocity was zero, I was left trying to compare 0 to 0.0508, which was impossible to do in the percent error calculations, so I decided to leave that one to a percent error of 51%, definitely a source of uncertainty. But overall, my calculated data pointed towards demonstrating conservation of momentum for both cases as a result.

In proving conservation of energy I had a similar process of writing out the energy of the system before and after the collision with the changing kinetic energies, but unlike momentum, energy is a scalar--thus I used the following equation to obtain the actual speed of the two masses, NOT velocities in the x and y directions.

Sample calculation of how I calculated these velocities for my energy equations.

In doing so, and calculating for the energy of the system, I observed that notable amounts of energy were lost:

1. I calculated the final and initial energies to not be close enough to one another for them to be considered equal. When the masses were equal, I got a percent error of approximately 53% for that first case. The initial energy, .42J, was much greater than the final energy, .20J, showing that a significant amount of energy was lost during the collision. 
2. This was less of a case on the second trial where the final energy, .00709J, was much more closer to the initial energy, .0095J with a percent error of 25%, showing that the loss of energy was not as significant in this case. However, it is nevertheless a relatively high degree of error.

Therefore, I could see that the total kinetic energy before the collision was not the same as total kinetic energy after the collision, and could therefore conclude that unlike momentum, in this collision, energy unfortunately was not conserved.

I obtained numerous graphs that show:

• Initial and final position of x for each of the balls for both cases
• Initial and final position of y for each of the balls for both cases
• Initial and final position of x AND y for each of the balls for both cases

2) Afterwards, I had to create graphs of the x&y centers of mass vs time for both position and velocity of the balls. Reason being, I concluded, that these graphs would show that the center mass of position and velocity would demonstrate the conservation of momentum and energy further; that the system behaves as if all of its mass were at the center mass.

To demonstrate this, I used the following equations to solve for the various center masses:

I entered these equations for xcm and ycm for both position and velocity in new calculated columns, and the resulting graphs demonstrate the original assumption: the center mass moves linearly, and its velocity of this point appears to be constant for BOTH cases.

Overall, I obtained numerous graphs that show:

• Xcm + Ycm for case 1
• Vxcm and Vycm for case 1
• Xcm + Ycm for case 2
• Vxcm and Vycm for case 2

Measured Data/Table:

Measured data of:
steel ball, aluminum ball, x-y components of velocity for case Equal Masses (table 1) and Different Masses (table 2)

Calculated Data:

Demonstrating the Conservation of Momentum in the X and Y directions for both cases.

Calculation of the Conservation of Momentum in the X and Y directions for both cases

Percent error of Conservation of Momentum

Calculating the Conservation of Energy for both cases
showing that energy is not conserved except for case 2 , with a percent error of 25%.

Sample of how I calculated center mass, in this case, for position in the Xcm. The same was done for Ycm, Vxcm, Vycm, and so forth.

Graphs:

Steel balls of equal Mass:

X vs Time for steel ball with initial velocity

Y vs Time for steel ball with initial velocity

X vs Time for steel ball with zero initial velocity

Y vs Time for steel ball with zero initial velocity

All of the above graphs overlayed into one.

Different Masses of steel and aluminum ball:

X vs Time for steel ball AND  aluminum ball

Yvs Time for steel ball AND  aluminum ball

X-Y Center Mass & V Center Mass for Equal Steel Masses

Xcm vs t and Ycm vs t overlayed

Vx-cm vs t & Vy-cm vs t overlayed

X-Y Center Mass & V Center Mass for Steel and Aluminum Balls

Xcm vs t and Ycm vs t overlayed

Vx-cm vs t & Vy-cm vs t overlayed

Conclusion:

Based upon my calculated and measured data, I can reasonably conclude that momentum was conserved due to how numerically close their initial and final values were, even if the percent error on one of them was 51% because of zero. Besides that, the percent error was very reasonable:

Percent Error for momentum

 However, the same could not be said for energy. Because of just how large the numerical difference, not just the percent error, of the initial and final energies of the cases were from each other. For us, energy cannot, at least for the two balls of equal mass, be said to have been conserved. There was a clear and significant loss of energy:

Percent Error for energy: Far right column

Uncertainty/Error

• We can determine a level of uncertainty in the measurement of the ball mass. The scales had an uncertainty of .01 grams, which would affect the second case calculations for momentum and energy more so than the first, as the variable mass simply cancels out when the two balls are of equal mass.
• Next, human inaccuracies (compounded by video quality) in positioning the balls in the position vs time graph would lead to an inaccurate slope; velocity, and therefore introduce a degree of uncertainty in our calculations for momentum AND energy.
• Most importantly, we also see that real-world collisions see losses of energy from a variety of sources: friction, sound, vibration, etc, all of which clearly affected our data--most significantly, our calculation of energy of the equal masses.