Lab 15: April 24,2017:Collisions In Two Dimensions

Lab 15 Collisions In Two Dimensions

Amy, Chris, and John

April 24,2017

In today's lab we watched two collisions take place and we were measuring if momentum and energy are conserved.

Theory: 

Fig. 1. The main apparatus in our lab.
Frictionless glass table where we perform the collisions.

Theory: 
Elastic collisions are both which the momentum and the kinetic energy are both conserved. The total energy system before the collision equals the total energy after the collision. If some energy were to be lost after the collision, then our definition would be inelastic collisions. Elastic collisions can also be conserved by solving it using the center-of-mass frame of reference. The frame of reference is from the perspective of the lab.

	Before:		After:
	x	y	x	y
m1	m1v1ix	m1v1iy	m1v1fx	m1v1fy
m2	m2v2ix	m2v2iy	m2v2fx	m2v2fy
Total	m1v1ix+ m2v2ix	m1v1iy+ m2v2iy	m1v1fx+ m2v2fx	m1v1fy+ m2v2fy

so from the table, the initial momentum in x will be equal to the final momentum in x and likewise for y:

To determine if energy was conserved, you test to see if the initial kinetic energy before the collision is equal to the final kinetic energy of both balls after colliding. Mathematically this looks like:

Note that there is no m2 times velocity 2 initial because our second mass is at rest before collision, hence velocity = 0. 

We will also be looking at graphs of position of the center of mass of both objects as well as the velocity at those locations. Our formula for center of mass calculations are:

      

 In our setup (shown in Fig. 1), after reorienting the axes to follow the trajectory of the first ball along the y-axis we expect the results for center of mass of x to show a constant zero (since it rolls along the x-axis) and the center of mass of y to steadily increase as the balls move further up in the y direction.

Apparatus and Procedure:

We have a frictionless glass table with a stand on the side that holds a camera. We chose to use my iphone since the slow motion feature can record up to 240 frames per second. A high frame rate allows us to get clearer pictures when we input data points in LoggerPro. 

The lab is split into two different scenarios, we chose to do:

1. Small glass marble colliding with another glass marble of equal mass.

2. Big glass marble of greater mass colliding with small glass marble (same mass as previous).

The lab procedure itself is rather simple, level the table to make sure the marble does not roll on its own, attach phone to the stand and record as we roll the other marble and make it collide with the stationary one. After getting two different recordings, we take our video to LoggerPro to extrapolate data from it via graphing data plots of the marble's movement. We can test the 2-D collision scenario and see if momentum and energy are conserved with the values we derive from our graphs. 

We use a LoggerPro feature of changing the axes on the graph so we can simplify calculations by moving the y axis onto the path of the first marble initially before collision. This gives us easier data to work with for calculations. To find values of velocity, we can find x and y components of both marbles before and after collision by finding the slope of the line on the position graph. 

For the center of mass graphs, we use the formulas laid out in the theory/introduction and introduce a new calculated column with this value. This allows us to plot these values vs time on a graph. We then used the derivative function within LoggerPro to derive the value of center of masses for x and y we obtained with respect to time. 

Data:

Our small marble mass was 4.8 g or 0.0048 kg in SI units. 

For the bigger marble the mass was 0.0193 kg.

For our first small marble on small marble collision:

The before collision speed of the marble. The value of the slope is given by the m component in the linear fit.

The velocity of the marble after collision. We see that slope (velocity) for Y1, Y2 and X1 are all roughly the same.

We see the velocity of X2 here after  collision. 

Velocity x and y, before and after:

	Before:		After:
	x	y	x	y
v1	0.39170	0	0.1505	0.1505
v2	0	0	-0.1404	0.1505

For the big ball colliding with the small (we actually oriented this one along X unlike the previous where we did it with Y). Note here that mass 1 is the bigger marble and mass 2 is the smaller:

The velocity of X2 after collision.

The velocity of Y2 after collision.

The velocity of X1 after collision. Notice it did not change too much from the initial.

For this portion of the lab we forgot to get a graph for the initial velocity of the bigger ball. Judging by the graph, The slope is steeper than the after collision value of 0.04083, so I will just be using the slope of X2 which looks like a good fit.

Velocity x and y, before and after:

	Before:		After:
	x	y	x	y
v1	0.05646	0	0.04083	0
v2	0	0	0.05646	-0.0106

Center of mass graphs:

Center of mass for x and y in the first scenario (small on small).

The center of mass velocity in x and y. 

Center of mass x and y for the second scenario (big on small).

The center of mass velocity in x and y.

Calculated Data:

So creating a table for each scenario similar to the one outlined in theory/introduction gives us:

Small on small scenario

	Before:		After:
	x	y	x	y
m1	0	0.0019	0.0007	0.0007
m2	0	0	-0.0007	0.0007
Total	0	0.0019	0	0.0014

Note: (I find v1 and v2 by using pythagorean's theorem sqrt([vx1]^2+[vx2]^2))

Big on small scenario

	Before:		After:
	x	y	x	y
m1	0.0011	0	0.0008	0
m2	0	0	0.0003	-0.0001
Total	0.0011	0	0.0011	-0.0001

KEi=12(0.0193)(0.05646)2=0.00003J

KEf=12(0.0193)(0.04083)2+12(0.0048)(0.0574)2=0.000024J

Conclusion:

Our goal of the lab to determine if momentum and energy is conserved and judging by the data we gathered and calculated, neither are conserved. Of course there are sources of uncertainty, poor data collection being one of them. There was also the fact that our initial assumptions aren't really correct. When objects collide, energy is lost through heat and our "frictionless" surface isn't truly frictionless. If there is any external force (friction for example in our lab) then momentum is not conserved. We also know that conservation of energy in collision only applies to perfectly elastic collisions. Even on a frictionless surface, intuition just tells us that the collision is not perfectly elastic (most things in nature are not). Heat produced by the collision of the two particles takes away from energy and disproves all our initial assumptions in the lab. In both cases, momentum final and kinetic energy final, the value obtained was less than what we initially started with, meaning energy was not conserved.