Lab 15: Two-Dimensional Collision

In reality, hardly anything occurs in one dimension. Luckily, the calculation for two-dimensional collisions are similar--it only requires that we consider the axises separately. That sounds like twice the work, and it would have been, if it weren't for the wonderfulness of technology. In this lab, we demonstrate the conservation of momentum, and calculate how much kinetic energy is lost, in a two-dimensional collision. And we do it twice to note the different behaviors depending on the conditions:

1. Lighter aluminum ball hits heavier steel ball
2. Heavy steel ball hits another steel ball of equal mass

We do this on a smooth glass surface, limiting the effects of friction and unpredictability of uneven surfaces. This surface is specially constructed for this, after all. Overhead is a video camera, the same one we used to capture coffee filters last month! We hit the ball in a glancing angle so that they go off into different directions, which makes calculating two-dimensional momentum more interesting. Unfortunately, since my pool skills are not up to par, I let my lab partner handle that, while I masterfully click the "Capture" button in Logger Pro.

There are only two of these set up around the room, so we borrowed this awesome looking USB flash drive to copy the recordings onto our own lab computers:

It's decapitated, ahh!! Before, we analyze the data in Logger Pro, first thing's first: We need the mass of our balls--no innuendo here. And no table scales either, since we need more accuracy. The balls keep rolling off the scale, so we use another weight to hold the ball in place.

The brass weight labeled '50 g' only weighs 48.5 g. My trust for lab equipment would have gone right out the window, but our room has no windows. Bad joke. Putting the steel and aluminum ball on the center hole of the weight, we were able to measure 115.5 g and 72 g, respectively. Subtracting the extraneous weight, the resulting weights are:

• Aluminum ball: 22.5 g
• Steel ball: 67 g

Armed with this miraculous data in hand, we are now ready to traverse the unknown... number of frames, one by one, carefully noting the position of the ball, from the initial flick to some time after the collision. By doing this, Logger Pro generates a graph with the x and y positions and velocity (by noting the distances between positions per frame). We can set an initial position, so Logger Pro knows how to offset the graph. We can also set a frame of reference, so Logger Pro could anchor the position to a known, real unit of measurement.

Our square glass plate measures 58 cm at one end, so we enter this into Logger Pro. And we end up with these two avant-garde, contemporary pieces of work which transcend notions of a single instance of time.

Using the position of each plot, Logger Pro generates two separate graphs, each with the x and y positions of each ball. In order to later solve the kinetic energy lost, we analyze, in Logger Pro, the slope of each ball before and after the collision. We'll leave that for now.

We could take a protractor to the angles of the trajectories in our capture above, or we could do some trigonometric calculations on the graphs, in order to find the x and y total momentum and total kinetic energy for each condition. Or, we could leverage technology to do the work for us. Choosing smart work over hard work, we create new calculated columns in Logger Pro. The total momentum per axis is just all the momentum summed, so we have:

p_x = m₁v_1x + m₁v_1x + m₂v_2x
p_y = m₁v_1y + m₁v_1y + m₂v_2y

And since kinetic energy is scalar, we use the Pythagorean Theorem for both balls, such that:

KE_TOTAL = 0.5 m₁(v_1x² + v_1y²) + 0.5 m2(v_2x² + v_2y²)

We plot these in a new graph against time, and get:

The initial slope of the results arise from Logger Pro interpolating and filling in gaps of data. Otherwise, it appears as if x and y momentum are relatively conserved. Other sources of error could be the imprecise nature of noting position from the captures; since the balls take up more than a single point, identifying the exact same point on the ball between the frames is difficult. The little bumps in the graph seem to reflect the little bumps in the capture trajectory. Kinetic energy is stable after the collision. Since we don't have enough data before the collision here, we turn back to the previous graphs with slopes to calculate the loss in kinetic energy. First, note that:

KE_loss = (KE_i - KE_f) / KE_i

Secondly, the slope of a position graph is the velocity. Dumping all the velocities into a spreadsheet table, we get:

Velocity	Aluminum Hits Steel	Steel Hits Steel
Cue Ball Initial X	0.482600	0.545900
Cue Ball Initial Y	0.000243	-0.075130
Cue Ball Initial	0.482600	0.551046
Cue Ball Final X	0.046430	0.159600
Cue Ball Final Y	-0.167400	-0.177400
Cue Ball Final	0.173720	0.238627
Object Ball Final X	0.109900	0.279600
Object Ball Final Y	0.045370	0.086380
Object Ball Final	0.118897	0.292639

 

And then, calculating for the kinetic energy:

Kinetic Energy	Aluminum Hits Steel	Steel Hits Steel
Total Initial	0.002620	0.010172
Cue Ball Final	0.000340	0.001908
Object Ball Final	0.000474	0.002869
Total Final	0.000813	0.004776
ΔKE	0.001807	0.005396
KELOSS	69.0%	53.0%

That's more energy loss than I expected. Nevertheless, we conclude that, in two-dimensional collisions, and perhaps regardless of any dimension, momentum is conserved, but not kinetic energy. Admittedly, it would be hard to conceive what mass and velocity would even mean in the context of a 9-dimensional collision. Maybe we shouldn't get too ahead of ourselves...