Collision in two dimensions

Objective:

• Observe two-dimensional collision and determine whether momentum and energy is conserved.

The experiment is divided into two parts:

1. Part 1: The collision between two steel balls
2. Part 2: The collision between steel ball and aluminum ball

Part 1: The collision between two steel balls.

Set-up:

• Let a glass square tripod sit on the table bench. Above it is a camera which is attached to a horizontal rod. 
• Use a C-clamp to secure the horizontal rod to the vertical rod, then connect the camera to a power supplier.
• Level the glass table so that if we let any object sit on it, that object won't move. The apparatus is shown below.

Figure 1: The apparatus of the experiment.

Set-up for the camera:

• Because the camera is very important in this experiment, we need to adjust it to make sure it will capture the collision between two steel balls correctly.
• In Logger Pro, we go to Video Capture Options, then choose Camera Settings, then Adjustments.
• In Adjustment, adjust the camera according to the figure below.

Figure 2: Adjusting the camera setting
• Then click on "Image", set Shutter to zero, set Exposure to Manual and reduce it, increase Gain until we are able to get a reasonable image from the camera.

Performing the experiment:

• First, we measure the mass of two steel balls. Since they are the same, we just need to measure one.

Figure 3: Measuring the mass of steel balls

m (steel ball) = 67.0g

• Then, we measure the length of one side of glass square in order to set scale later when analyzing the video.

Figure 4: Measuring the length of one side of the glass square.

L = 58cm ± 0.1cm

• Let a steel ball stay still in the middle of the glass square. Aim the rolling ball so that it hits the side of the stationary ball.
• The ball should ideally roll of at some decent angle from one another.

Prediction:

• Since the glass square is assumed to be frictionless, which means no external forces exert on the system, thus we predict that the momentum before and after collision should be equal each other. 
• We also predict that the energy is not conserved in this experiment because some of initial kinetic energy is transferred to internal energy during collision, thus the total kinetic energy of both balls after collision would be less than that of initial kinetic energy. Also, the collision is not perfectly elastic, therefore the conservation of energy is not applied.

Data Collection and Analysis

• First, we capture the collision of two balls. 

Figure 5: Dotting positions of both balls before and after collision.

• Using Logger Pro, analyzing the video by dotting positions of both balls, setting scale and origin. When we finish, we obtain the results shown below.

Figure 6: Those graphs of position vs. time after analyzing the video.

• Then we set up New Calculated Columns in Logger Pro to calculate initial momentum, final momentum, initial kinetic energy and final kinetic energy. For momentum, we need to consider two components: one in x-axis and the other in y-axis.
• The momentum of steel ball in x-axis can be found by multiplying mass of the steel ball with the x velocity

• The momentum of steel ball in y-axis can be found by multiplying mass of the steel ball with the y velocity.

• The total momentum in x-axis is found by summing the momentum of two balls in this direction, same for y-axis.






• The initial kinetic energy can be found by using an equation below
  
• The final kinetic energy can be found using an equation below

• After setting up those equations above in Logger Pro, we obtain the following result table.

Figure 7: The result table.

• Since we calculated the total momentum in x-axis and y-axis, we just need to graph them to see whether they are constant or not.

Figure 8: Graph of total momentum in x-axis and y-axis

• From the graph, we can see that the values of total momentum in x-axis and y-axis are more or less almost constant, which implies that the momentum is conserved in this collision
• We also analyze and graph initial and final kinetic energy to check whether it follows our prediction.

Figure 9: Graph of initial and final kinetic energy.

• As you can see from the graph, the mean of initial kinetic energy is 0.008068J, which larger than final kinetic energy (0.003987J). This means that part of initial kinetic energy is lost during collision like prediction. And the kinetic energy loss is calculated like below.

Part 2: The collision between steel ball and aluminum ball.

Set-up:

• We would use the same set-up as part 1 except replacing the stationary steel ball with an aluminum ball. Since we use a new ball, we need to measure the mass of this ball.
• The camera settings keeps the same as well, and we would follow the same procedure in part 1.

Prediction:

• Because we keep the same apparatus, the system doesn't have any external forces, thus the conservation of momentum is still applied. However, the mass of aluminum ball is smaller than that of steel ball, it would lower the value of momentum.
• The energy isn't conserved though. During the collision, an uncertain amount of kinetic energy is transferred to internal energy, thus those kinetic energy left is not equal to initial kinetic energy.

Data Collection and Analysis:

• Measuring the mass of aluminum balls, since we already have the mass of steel ball from part 1

Figure 10: Measuring the mass of aluminum ball

m = 10.0g

• Then capturing the collision of two balls.

Figure 11: Capturing the collision of two balls

• Analyzing the video to obtain the graph like part 1

Figure 12: Graph of position vs. time

• Setting up those equations used in part 1 in Logger Pro again to calculate initial and final momentum in x-axis, y-axis, then total momentum in both axes, lastly initial and final kinetic energy. When we finish, we obtain the result table.

Figure 13: Result table for part 2.

• We just graph total momentum in x-axis and y-axis like part 1 to check our prediction.

Figure 14: Graph of total momentum in x-axis and y-axis

• As we can see from the graph, the values of total momentum in both axes are more or less almost constant. Therefore, we can conclude that the conservation of momentum did apply in this collision.
• Then, we graph initial and final kinetic energy to see whether it's conserved or not.

Figure 15: Graph of initial and final kinetic energy.

• From the graph, we obtain the mean of initial and final kinetic energy is 0.008603J and 0.005745J, respectively. The value of initial kinetic energy is higher than that of final kinetic energy, which implies that there is a loss of energy during collision. This energy loss is calculated like below.

Discussion of result:

• Two significant things in both parts are the "close" conservation of momentum and a big loss of energy during collision

• First, momentum is not perfectly conserved because:
• The glass table is not ideally frictionless.
• There is an uncertainty in measuring the length of glass square which may affect the scale when analyzing the video.
• When we analyze the video, we didn't get the center of the ball every time, thus it may alter the velocity as well as position of each ball.
• Secondly, the big loss of energy may be caused by:
• It's not elastic collision, so the rolling ball may "stick" to the stationary ball a little bit before pouncing at different angle. 
• The inconsistency in pinpoint the center of mass of the ball affects the value of velocity, thus affects the value of kinetic energy.

Conclusion:

• In this lab, we perform the collision between two steel balls, and between one steel ball and aluminum balls. The purpose of both parts is to check whether the conservation of momentum and energy is applied in such this collision. 
• In part 1, if we ignore some errors that explained in discussion part, the momentum is conserved. Meanwhile, the conservation of energy isn't applied because this is not perfectly elastic collision. Some parts of initial kinetic energy is transferred into internal energy. The energy loss we calculated in part 1 is 50.58% which is very high. However, we come up with some reasonable errors that compensate that loss (explanation in discussion part)
• In part 2, the momentum is still conserved, while the energy is not. The energy loss in part 2 is 33.22%, which is better than part 1 but it's still pretty high.