Apr 22, 2015: Lab 15: Collisions in Two Dimensions

Purpose:

We want to see if there is a conservation of momentum in 2D, by looking at both same mass and different mass collisions.

Experiment:
Set-up:
- The camera apparatus is already set up so we only have to change some video capture settings.

- We want to keep the default settings of camera recording 25 second video. On LoggerPro, go to "Video capture options", "camera settings", "adjustments", "image". Adjust "shutter" to zero and saturation as shown below or higher, so each captured frames could be clearly analyzed in the next part.

- Make sure that the glass table is leveled. Test this by making sure the balls stay stationary and doesn't roll, this means the glass table is leveled.
-Measure two balls of different masses. Measure the glass table (without frame).
- Place two balls with the same mass gently on the glass frame. One at the the middle of the frame.
- Roll that other ball toward the one in the center so that when they collide, they roll of at some good angles from each other.
- Repeat the same steps with two balls of significantly different masses.
- The video of collision should take between ~ 1 to 2 seconds.

Data and Analysis:

glass length (for set scale) = 0.585m

M (steel ball)= 0.067kg; m(aluminum)= 0.01kg

Same mass collision: 2 steel balls in this case
- At this point, we have 2 good videos of 2 collisions.
- For each video, on the right side, click "set scale" and set the length of the glass measured from above. "Set origin"; choose the x-axis to align with the impactor's path.

same mass collision

-Then "add points series" for the impactor's position (the ball you rolled) for X and Y. Then, add another point series for the ball that got hit. X and Y, as shown above.
- Note that as you add points for the ball that got hit, add points at the same time frame with impactor ball. Therefore, before collision, the ball in the center would be at rest and the position shall remain the same before collision.

-Correlations (from linear fit) from X, Y, X2, and Y2 positions should be close to one in same mass collision.

- After obtaining 3 points series, use these in "new calculated columns" for:

Impactor's momentum (x)= M*(x-velocity)^2

Impactors' momentum (y)= M*(y-velocity)^2

Hit momentum (x)= M*(x-velocity2)^2

Hit momentum (y)= M*(y-velocity2)^2

Total momentum (x)= Impactor's momentum (x) + Hit momentum (x)

Total momentum (y)= momentum (y) + Hit momentum (y)

KE-impactor= (0.5)M(x-velocity^2+ y-velocity)^2

KE-centered ball= (0.5)M((x-velocity2)^2+ (y-velocity2)^2)

* Using Pythagorean theorem, V= sqrt [(Vx)^2 +(Vy)^2]. Plugging this into KE=(1/2)mV^2= (1/2)m[(Vx)^2 +(Vy)^2]

KE-initial = KE-impactor

- Since we want to see whether x and y momentum of this system is conserved, what we expect from graphing the momentum are relatively straight lines. The correlation of 0.9 for total momentum (x) tells us that ~ 90% of momentum in this axis is conserved, while only ~ 63% is in the y-axis.

- Also, note that both momentum graphs shown below, slightly decrease with time.

-As it is a nearly perfect elastic collision, we can also prove that kinetic energy of both axis are conserved. When the first is given a push, it's at the maximum kinetic energy. As collision occurs, M1's KE climbs down to 0.010J and M2's KE jumps from zero to 0.001J. After collision, both masses appear to share similar KE of 0.005, as shown in the continuous straight line. So, if we were to take a point from both masses' KE, both will add up 0.010J (energy of mass 1 right before collision).

- We could also prove the conservation of momentum and kinetic energy through calculations,

- Vo is the velocity right before collision, meaning right before the centered ball gains some velocity. V1 and V2 are velocities right after collision. And since there are more significant changes in position in x-axis, our error is therefore much smaller.

Different mass collision: "m" hitting "M":

graph of position X, Y, X1, Y1; Linear fits are not really necessary here, poor correlations

- With a smaller mass hit a larger one, we can use the same methods to prove conservation of momentum and kinetic energy.

- Proving conservations of momentum and KE through calculation,

- Kinetic energy of "m" and "M" seem to be at 0.005J. After 1.2 seconds, KE of "M" drops a bit, while kinetic energy of "m" stay relatively at 0.005J. This suggests an external work (thermal energy) acting on "M".

KE-initial= Impactor's KE; KE-final= M2's KE

- In momentum graph blows, the correlation of 0.66 suggests that only 66% of total momentum is conserved. Likewise, we can look to this more in detail by getting correlation of total momentum (x) and total momentum (y). Not shown here, but the correlations are: 0.72 and 0.48, respectively. Once again, we see that more momentum in x-direction is conserved.

- If we compare both KE and momentum's graphs of this experiment to the last one, there's more uncertainty and error in this one since the data points collected from mass M, doesn't show significant changes in position.

Conclusion:

When it comes to collecting data, the video camera gives more accurate results with significant changes in position of the balls. The fact there are some drop frames and that the camera doesn't show  a flat picture for analysis also plays a role in our systematic error. Other usual errors are from mass, length measurements. Nonetheless, within our uncertainty, we conclude that there are conservation of momentum and conservation of kinetic energy in two dimensional collisions.