26-Apr-2017: Lab 15: Collisions in Two Dimensions

Lab 15: Collisions in Two Dimensions

Joel Cook

Nina Song

Eric Chong

Lynel Ornedo

Lab performed April 24, 2017

In this lab, we are attempting to determine if momentum and energy are conserved in elastic collisions between objects. 

Intro: When two similar objects collide in two dimensions we expect that momentum and energy will be conserved from before and after the collision. To verify this, we will use a level, glass table and collide spherical objects. By capturing the collision in slow motion, we can analyze the video of the collision to gather data to be used to calculate momentum and energy.

Procedure: The glass table mentioned previously is shown below.

The table is leveled before the experiment such that a steel ball placed in the middle will not roll. The table has a stand attached that will hold a smartphone to facilitate filming in slow motion. The width of the table was measured with a meter stick to be used in scaling the video during analysis.

For the first trial of the experiment, a steel ball was placed in the center of the table and another steel ball (of equal mass and size) was rolled towards it. The trial was repeated until the balls collided in such a way that the two balls travelled away from each other after the collision. The slow motion video was captured on a smartphone and transferred to a laptop to use Logger Pro for analyzing the video. 

For the second trial of the experiment, a glass ball (of lower mass) was placed in the middle of the table and a steel ball was rolled toward the glass ball, as before. 

From the video analysis, velocities were recorded for before and after the collision to compare momentum and energy to establish if the quantities were conserved. In addition to velocity vs time and position vs time, graphs were created for x and y center of mass vs time and velocity of x and y center of mass vs time.

Data and Analysis:

Steel v Steel:

The graph below shows the x and y positions vs time of the two steel balls from before until after the collision. X2 (or Y2) is the ball that was at rest in the center of the table top.

The initial and final velocities for the x and y directions were calculated from the slopes of the position vs time graph shown above. The data is in the tables below. The first table is for the steel ball initially rolling and the second table for the ball that was initially at rest.

Velocity (m/s)

Xo	Yo	Xf	Yf
0.007914	-0.3435	0.1100	-0.1258

Velocity (m/s)

X2o	Y2o	X2f	Y2f
0.005618	0.001200	-0.07704	-0.1635

As seen in the graph, the two steel balls have a constant velocity before the collision and travel away from each other at equal and opposite velocities.

We expect the momentum to be conserved for the collision.

po = pf

mv1o + mv2o = mv1f + mv2f

Masses are equal and therefore cancel.

X direction:

0 + 0 = (0.1100) + (-0.07704)

0.1100 = 0.07704

A difference of approximately 30%.

Y direction:

(-0.3435) + 0 = (-0.1258) + (-0.1635)

-0.3423 = -0.2893

A difference of approximately 16%.

The kinetic energy of the system should be conserved, as well.

KEo = KEf

0.5*m*(v1o)^2 + 0.5*m*(v2o)^2 = 0.5*m*(v1f)^2 + 0.5*m*(v2f)^2

(The velocities used were calculated by taking the square root of x-velocity squared plus the y-velocity squared.)

Masses are equal and therefore cancel.

0.3459^2 + 0.00574^2 = 0.167^2 + 0.181^2

0.1197 = 0.06065

A difference of approximately 50%.

The next set of data calculated was the center of mass of the system for the x and y directions. The following photo shows the position of X, X2 and Xcm for the collision.

Since there is no acceleration of the center of mass of the system, we expect the speed of the center of mass (as calculated from the slope of the position v time graph) to be constant. As shown in the photo, the velocity of the center of mass before the collision is 0.008957 m/s and after is 0.01676 m/s. This is a difference of approximately 50%.

The following photo shows the position of Y, Y2 and Ycm for the collision.

As shown in the photo, the velocity of the center of mass before the collision is -0.1698 m/s and after is -0.1442 m/s. This is a difference of approximately 15%.

Finally, the velocity of the center of mass was graphed with the velocity of each ball for the x and y direction.

The next photo shows the Xcm.

The slope of the velocity of the center of mass vs time will give us the acceleration of the center of mass. We expect the acceleration to be zero. As shown, the acceleration of the Xcm for the system is 0.02077 m/s^2. 

The next photo shows the Ycm.

As shown, the acceleration of the Ycm for the system is 0.1022 m/s^2. 

Steel v Glass:

The graph below shows the position vs time data for the steel ball (X/Y) colliding with a glass ball (X2/Y2) of lower mass. The glass ball was in the middle of the table and the steel ball was rolled toward it.

The initial and final velocities for the x and y directions were calculated from the slopes of the position vs time graph shown above. The data is in the tables below. The first table is for the steel ball initially rolling and the second table for the ball that was initially at rest.

Velocity (m/s)

Xo	Yo	Xf	Yf
0.03314	-0.5269	0.09021	-0.1817

Velocity (m/s)

X2o	Y2o	X2f	Y2f
-0.0003901	0.0003353	-0.07887	-0.4257

X direction:

po = pf

mv1o + mv2o = mv1f + mv2f

(0.029)*0.03314 + 0 = (0.029)*(0.09021) + (0.020)*(-0.07887)

0.000096 = 0.00103

A difference of approximately 7%.

Y direction:

(0.029)*(-0.5269) + (0) = (0.029)*(-0.1817) + (0.020)*(-0.4257)

-0.01528 = -0.01432

A difference of approximately 6%.

Kinetic energy was compared before and after the collision.

KEo = KEf

0.2787 = 0.2286

A difference of approximately 18%.

The following photo shows the position of X, X2 and Xcm for the collision.

As shown in the photo, the velocity of the center of mass before the collision is 0.01764 m/s and after is 0.01364 m/s. This is a difference of approximately 23%.

The following photo shows the position of Y, Y2 and Ycm for the collision.

As shown in the photo, the velocity of the center of mass before the collision is -0.2817 m/s and after is -0.2952 m/s. This is a difference of approximately 5%.

Finally, the velocity of the center of mass was graphed with the velocity of each ball for the x and y direction.

The next photo shows the Xcm.

As shown, the acceleration of the Xcm for the system is -0.01965 m/s^2.

The next photo shows the Ycm.

As shown, the acceleration of the Ycm for the system is 0.04071 m/s^2.

Conclusion:

Our data and analysis suggests that the momentum and energy were closer to being conserved in the glass vs steel ball collision than in the steel vs steel ball collision. Some of the difference between pre-collision and post-collision were quite large. A few things could have affected our results. First, the video analysis relies on clicking accurately and consistently on the center of the balls in a less than crystal clear video. This will affect the accuracy of the data and impact the comparison of momentum and energy. Second, we are assuming in our lab that the balls are rolling and never sliding. Clearly, this is not accurate because it is likely that the one of the balls is sliding after the collision before it begins rolling. Therefore, some of the energy is going to kinetic friction. Third, the speeds which we are analyzing are very small. Differences of a few centimeters per second are magnified at these scales. Lastly, it is not possible to have perfect initial conditions. It is possible that our table top was not perfectly level or that the any debris on the surface would affect the rolling motion of the objects. These impacts should be fairly minor.

Overall, the results of our lab were mixed. It seems reasonable that the glass vs steel collision was successful, as the results suggested more certainly that momentum and energy were conserved. It is likely that a more careful repeat of the steel vs steel collision would yield more acceptable results that would show momentum and energy being conserved.