22-April-2015: Collisions in Two Dimensions

The purpose of this lab was to determine whether not momentum and energy are conserved in a two-dimensional collision.

Set Up:

Intro:

To perform this lab, we needed two create an environment in which we could create, observe, and analyze a two dimensional collision. The collision would occur between two marble in two separate trials. The first would consist of two marbles with similar masses and the second would consist of two marble with different masses. To create a reasonable surface to conduct the lab, we used a leveled glass table. The idea was that we would leave one marble stationary at the center of the leveled glass table and another marble would be aimed directly at the stationary marble. Once ready to begin collecting data, we would flick/push our marble towards the stationary marble. Ideally, the balls would deflect off each other at some angle from one another. Theoretically, we should be able to conclude whether not momentum in the x- and y-directions was conserved and whether or not energy was conserved.

Actual Apparatus:

We setup the experiment just as describe above, with a leveled glass table and two marbles of similar mass and two of different. To observe and collect data from the collision, we had a camera setup directly above the leveled glass table. This camera would be hooked up to LoggerPro and would allow us to split up the velocity of the marbles into x- and y-components. This would in turn allow us to find the momentum in the x- and y-directions (Fig. 1)

Fig. 1
With one marble stationary, we would flick another marble at it and record the collision with our LoggerPro camera.

Part 1- Two Marbles of Similar Mass:

Fig. 2
When setting our origin, we tried to line up at least one axis with the path
of the flicked marble. This would leave us with only one value of velocity to
calculate in our initial stages before the collision. 

To begin the lab, we first  used the apparatus with two marbles of similar masses. The first marble, which we will call marble one, was measured to be 0.021 kg. The second marble, which we called marble 2, was measured to be 0.019 kg. We could conclude that these two marbles had relatively similar masses. As stated in the intro, we needed to flick one marble at the other while recording the collision. Once we were ready, we began collecting data and performed the collision. We analyzed the video capture in LoggerPro (see Fig. 2).

Fig. 3
This position graph shows us the position of the two marbles
 in both the x- and y-directions.

To analyze the video, we needed to set a scale and origin for the collision. Our scale was placed in the top right corner of the glass table, where we measured a small piece of it to be roughly 22.3 cm. We placed the origin at the stationary marble and attempted to line up the y-vertical axis to the path the flicked marble took. We then placed frame by frame dots at the positions of the marbles before and after they collided. We flicked marble one at marble two.

Once we analyzed the video, we were given enough data two obtain the graphs of velocity and position in both the x- and y-positions for the two marbles (see Fig. 3). For our trial, we would be mainly using the position versus time graphs. To find the momentum and energy of the marble, we needed to find the velocities before and after the collision in both the x- and y-directions. To do this we simply took the slope of the position graphs (see Fig. 4).

Fig. 4
If we found the slope of the position graphs, we could find our velocity in the x- and y-directions for both of the marbles.

Once we had the velocity of the two marbles, it was now time to manually calculate the change in momentum and kinetic energy of the two marbles. For momentum, we broke the problem into x- and y- components (see Fig. 5). For kinetic energy, we simply found the resultant velocity of the x- and y- velocities and used this value in the equation KE=(1/2)*m*v^2. (see Fig. 6).

Fig. 5
Calculations for the initial and final momentum. We lost 28.6 % of our initial momentum after the collision.

Fig. 6
 Calculations for the initial and final energies. We lost 64.7 % of our kinetic energy after the collision. 

Once we had our initial and final values, we compared our values for initial and final momentum and energy. In both instances, we wounded up with less in the final stage, which is reasonable given the fact that there are more forces acting on this system than we are accounting for. This will be further discussed in the error section.

Part 2- Two Marble of Different Masses:
The second part of the lab included the same apparatus, only this time the two marbles would have different masses. We would flick a small marble (0.005 kg) at our original marble one (0.021 kg). Once again, we captured the collision on video and analyzed it on Logger Pro.

Just like before, we set our origin, scale and points on the video. This in turn would allow us to analyze the velocity of the two marbles. Just like before, we were planning to use the slope of the position graphs to find our initial and final momentum and energy (see Fig. 7)

Fig. 7
Again, we found the slope of the position graphs to find our values for velocity.

Once we had our values for velocity in the x- and y-directions for both of the marbles, we then found the initial and final momentum and kinetic energy for the system (see Fig. 8)

Fig. 8
Once we had our values for velocity, we calculated our momentum and kinetic energy for the collision.

Again, we found that we had lost some energy and momentum in our collision, however, this may have been due to the error in our lab.

Conclusion and Error:

As we can see from the collected data, energy and momentum was not perfectly conserved in this collision. This may have been due to the following sources of error in our lab:

• The surface of the glass table and marble were not completely frictionless, which suggests the presence of an external force.
• As we can see Fig. 2, the video capture did not give us an accurate image of the collision. Instead, it was slightly curved. This may have altered our ability to find the true position of the marbles. 
• Again with the video capture, the marble moves quickly enough to make it difficult to find its pin point center in every frame, causing there to be some uncertainty in its position.
• All of these values become estimates as the sources of error pile on.

However, although we had all these sources of error, are results are relatively realistic. In every instance, the difference in energy and momentum is a loss, which in a real life situation is probable. Overall, in a perfect world this lab would have yielded us a more accurate depiction of the conservation of momentum and energy.