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...

Lab 14: Impulse-Momentum Theorem

In a collision, kinetic energy is often transferred into heat and sound energy, such that it is hard to measure and not conserved. In such cases, we use conservation of momentum to model the reaction of the collision. Momentum is defined as:

p = mass * velocity

We also know that momentum is caused by force over the time interval which the force acts. We call this impulse, which is defined as:

J = Force * time

Therefore, since the momentum of the system cannot change without forces, we know that, if friction and gravity could be disregarded, since the magnitude of force before and after a collision must be 0, there is no change in momentum before and after the collision. And unlike kinetic energy which could be transferred into other energy forms, momentum, given enough time, only transfers momentum to other objects. Thus, we say that momentum of a system is conserved, regardless of elastic or inelastic collisions (which just vary in force-time). This is shown by the Impulse-Momentum equation (the change of impulse is equal to the change of momentum):

F * Δt = m * Δv

...wherein the change in momentum mΔv could also be defined as:

m * (v₂ - v₁)

If we know the force and velocity of the moving object, then we could test whether this is true. But how is this possible?!

Once again, we produce our trusty gadgets: the motion sensor and the force sensor! As pictured above, we have two wheeled carts on a level track, making the effects of friction and gravity (hopefully) negligible. We put pieces of paper under the track until it is precisely leveled, showing less than 0.4° on the phone app, such that the cart does not roll on the track by itself. Then we mount a force sensor atop it with a rubber attachment to simulate an elastic collision. The rubber would bounce off the translucent plastic on the second cart, causing force over the collision time. We measure the cart with the force sensor to be m = 433 g. The motion sensor, attached to the other end of the track, measures the velocity and position of the track.

So... we hook the sensors up to Logger Pro, and give the red cart a push, being careful the cables don't contribute tension or get in the way of the sensors. Logger Pro gives us the standard force vs time, and velocity vs time graphs:

What we're interested in here is the change of impulse, which can be obtained by using the integral function to find the curve under the force curve during the collision. On the other hand, we find the change of momentum by finding the highest and lowest velocities right before and after the collision, and multiplying that by the mass. In our first experiment, we find that our impulse is 0.5358 Ns. Our Δv, as reported by Logger Pro analysis, is 1.282 m/s. Multiplied by our cart's mass of 0.433 kg, we get the figure calculated in purple in our data table. Although it reads impulse, it is actually momentum (but also impulse!), coming in at 0.555 kg m/s. This is under 4% error, a relative success by our standards--it shows a clear correlation.

The instructions require us to run a second experiment with additional mass, and a third inelastic collision without the mass, but having messed up the inelastic collision once, we ended up doing it with the mass on. It should be inconsequential. The higher mass supposedly demonstrates that larger momentum change is conserved, while the inelastic collision increases collision time. But by the time we got to the second experiment, we've realized that the force sensor has less chance of peaking if we pushed the cart slower, therefore our second and third experiments actually showed less velocity, and therefore less momentum change.

Without further ado, we've tied a 200 g mass to the back of the cart for our next two experiments, bringing the total mass of our cart up to 633 g. Our second push yielded:

This one was pushed much slower to ensure that the force sensor captures without problem, but also to ensure that the mass stays in place. As such, the Δv is only 0.479 m/s; multiplied by 0.633 kg, the resultant change in momentum is approximately 0.303 kg m/s. Using integral again, our impulse is 0.2994 Ns. This time there's less than 2% error, not bad!

The third trial, once again, is with the additional mass, and also with a nail attachment to the force sensor, and putty on the other end, in order to create an inelastic collision. The capture looks like this:

The cart was pushed faster, but ended up stuck to the putty at 0 velocity, so the change in velocity turned out about the same as the second trial, at Δv = 0.510 m/s. Once again, using 0.633 kg mass, we calculate 0.323 kg m/s momentum change. The impulse turned out to be 0.3210 Ns. Under 1% error.

The point here is that, regardless of the mass or material of the cart, momentum and impulse are conserved, thus proving that the Impulse-Momentum Theorem is true. One unexpected result from this experiment is that the collision time are all under 1/5 of a second, and neither mass, nor the putty, seemed to greatly affect it.

Lab 13: Magnetic Potential Energy

Thus far, we have stuck to energy forms with neat, predefined formulas to calculate using the work-energy theorem. In fact, all kinds of energy can be used in that way. This time, we demonstrate that this is true with magnetic potential energy, whose formula we derive. We do this by experimentally recording magnetic force under different circumstances. Magnetic force which repels two objects will have magnetic potential energy, since the objects, which have the inclination to move towards each other, is being forced apart by a distance. As before, we graph this relation between force and distance and integrate to find the work caused by the energy.

To do this successful, we must first reasonably eliminate friction. Pictured above is an air track with little holes connected to a blower, which pushes air through those holes. The red cart on the air track will approach 0 friction, such that it could be slid across the track unimpeded until it hits the end. At the ends of the air track and the cart are magnets which repel each other, causing the cart to bounce elastically from the end even if the magnitude of velocity overcomes the repelling force. However, first, we'd like to discover the relation between this magnetic force and distance, so to alter the role of gravity, we lift the air track to different angles and record how far the cart is held from the ends. The force of the magnet is, of course, mg sinθ. We label distance x, in meters.

The air track is lifted by placing a number of books under the other end. This rudimentary method to change the angle is counteracted by the advanced cellphone angle measuring apps that put Inspector Gadget out of business. The measured angles should be accurate relative to each other, but we expect an offset due to the geometry of the phone, and that the table isn't exactly flat. There is a ruler taped to the end of the air track, wherein the position of the magnet measured at 527 mm. This aided in measuring the distance, for all we had to do was measure the position of the magnet on the cart in the same way, and find the difference. We recorded 8 data points, although the first or last may not be as accurate, as we later found. The data is as follows:
 

Position (mm)	x (m)	θ (º)	sinθ
476	0.051	1.1	0.0192
487	0.040	2.7	0.0471
492	0.035	4.9	0.0854
497	0.030	6.9	0.1201
500	0.027	8.8	0.1530
502	0.025	10.5	0.1822
503	0.024	12.1	0.2096
506	0.021	14.0	0.2419

 Now that we've gathered the data, we can plot the graph to find the relationship:

We set a curved fit that we expect the data to conform to. The data doesn't exactly fit, but it is reasonably close considering the experiment. Here, we've found our magnetic force equation to be F = 3.168 x 10^-5 r^-2.690. We integrate and flip the signs to get the potential energy:

We can now test our magnetic potential energy against kinetic energy to see whether energy is conserved like we expect it to. If our calculations are close, then we should see the pattern of conservation. To do so, we attach a motion sensor to the end of the air track and measure its offsets from the magnet. We need to know the differences to accurately reflect distance. We also make sure the air track is horizontally flat, such that the cart doesn't move due to gravity.

We measure that the back plate or our cart, which the infra-red beam from the motion sensor would hit, is at position 433 mm. Thus, if we subtract it from the end position at 527 mm, so have 94 mm. This figure will be entered into Logger Pro to calibrate the motion sensor.

The thinking is this: If we give the cart a push, then record its movement with the motion sensor, we should get its position (distance) and velocity, which should be enough to calculate the potential energy and kinetic energy, respectively. Entering new calculated columns:

And summing up total energy, we get:

The cart hitting the end of the air track causes the dip in kinetic energy, and at the same time increases magnetic potential energy since the magnets approach each other. We can see that the energies are inverse of each other, causing the sum to be relatively even compared to the total energy in the middle of the track. The larger spikes in energy can be explained by gaps in the motion capture, causing Logger Pro to interpolate and guess what's there. The smaller spikes are likely explained by the previous inaccurate modeling of the force, possibly due to loose measurements of the angles. Also, despite the air track, friction isn't completely eliminated, nor air resistance.

Here is a snapshot of our data, exported to a CSV file, with the gaps taken out (which have 0.001 J kinetic energy). As you can see, the total energy is even.

Time	Position	Velocity	Acceleration	Kinetic Energy	Potential Energy	Total Energy
1.65	0.625	-0.268	0.109	0.013	0	0.013
1.7	0.608	-0.252	0.216	0.011	0	0.011
1.8	0.587	-0.248	-0.111	0.011	0	0.011
1.85	0.574	-0.254	-0.067	0.011	0	0.012
1.9	0.561	-0.255	0.008	0.011	0	0.012
1.95	0.548	-0.252	0.058	0.011	0	0.011
2	0.536	-0.247	0.058	0.011	0	0.011
2.1	0.513	-0.251	-0.096	0.011	0	0.011
2.15	0.498	-0.259	-0.011	0.012	0	0.012
2.2	0.486	-0.251	0.059	0.011	0	0.012
2.25	0.473	-0.248	0.046	0.011	0.001	0.011
2.3	0.461	-0.247	0.029	0.011	0.001	0.011
2.35	0.449	-0.246	0.027	0.011	0.001	0.011
2.4	0.437	-0.245	0.041	0.01	0.001	0.011
2.45	0.424	-0.243	0.071	0.01	0.001	0.012
2.5	0.412	-0.238	0.117	0.01	0.001	0.011
2.55	0.4	-0.232	0.187	0.009	0.002	0.011
2.6	0.389	-0.221	0.317	0.009	0.003	0.011
2.65	0.378	-0.205	0.573	0.007	0.004	0.012
2.7	0.368	-0.172	0.999	0.005	0.007	0.012
2.75	0.359	-0.108	1.461	0.002	0.011	0.014
2.85	0.358	0.08	1.573	0.001	0.013	0.014
2.9	0.365	0.154	1.15	0.004	0.008	0.012
2.95	0.374	0.196	0.677	0.007	0.005	0.012
3	0.385	0.215	0.332	0.008	0.003	0.011
3.05	0.396	0.221	0.168	0.009	0.002	0.011
3.1	0.407	0.227	0.112	0.009	0.002	0.011
3.15	0.419	0.232	0.076	0.009	0.001	0.011
3.2	0.431	0.235	0.043	0.009	0.001	0.011
3.25	0.443	0.236	0.018	0.009	0.001	0.011
3.3	0.454	0.236	0.007	0.009	0.001	0.011
3.35	0.466	0.237	-0.007	0.009	0.001	0.011
3.4	0.478	0.235	-0.01	0.008	0.001	0.011

Lab 12: Conservation of Energy

An oscillating mass-spring is a relatively closed system, so the total energy should be preserved even if kinetic and potential energies alternate. For this lab, we look at how to use calculus to derive a formula for all the energies in play, and then measure them in Logger Pro to see that total energy has, in fact, been preserved!

Although the math is long, we will discuss it a little bit to provide a context. The above diagram shows a spring of arbitrary length L, hanging on some apparatus at height h above ground. The mass of the spring weighs to be m = .100 kg. When we do physics problems, we tend to represent these object with a single point, but in real life the spring takes up quite a bit of space. Therefore, since gravitational potential energy depends on height, the potential energy of the spring actually varies from top to bottom. To solve this problem, we see that it is theoretically possible to slice this spring horizontally into a number of pieces, each with a portion of the mass of the entire spring, we call dm. Each dm slice is also the overall mass m divided by overall length L, times the height of the slice, or change in height dy. We can sum up all the slices to get an approximation of potential energy, with it being more accurate as the slices approach infinity, such that:

Note that this is the same as integrating, which we can do to solve for gravitation potential energy:

In class, we had derived the kinetic energy in this manner as well, but for the sake of brevity (and my precious time), I present to you the formulas. Ta-da!

Now, the set up is easy: We hang a lengthy weighted (weights are .600 kg) spring on a stand, over a motion sensor on the ground. The motion sensor is, of course, hooked up to our trusty Macbook running Logger Pro, so that we could record the position, and the corresponding velocity, of the spring, by pulling it and letting it go. The hardest part was aligning the motion detector, and making sure the spring stays over it, and making sure the weights don't fall and damage the motion detector, causing a lawsuit!

What was really helpful was taping a piece of paper under the weights so that it makes it easier for the motion sensor to pick it up. Here's out data:

All those wavy lines... The only thing missing now is the spring constant k, and the unstretched height of the spring. Although we installed a force sensor to our spring, we didn't actually use it. Finding k was simpler measuring the spring at rest without any mass (96 cm from ground), and the spring at rest with the mass (72 cm from ground), and finding the difference, such that:

kx = m_weightsg

k = m_weightsg / x

k = (0.6 kg * 9.8 m/s²) / 0.24 m
k = 24.5 N/m

We also do a run in Logger Pro with the spring at rest with mass, just to see how far the motion detector figures that height to be. Since we didn't zero or calibrate the motion sensor, there is expected to be an offset. This will help us find elastic potential energy, which requires that we find the stretch of the spring. With all the variables known, we start making calculated columns in Logger Pro with the formulas we've discovered, as shown above. We substitute y for the position data, and v for the velocity data. Once completed, we sum up the potential energies of the mass and spring, both elastic and gravity. We also sum up the kinetic energy of the mass and spring. Finally, we get the total between the two as well. If theory stands, this final total should be relatively stable, since all the variable energy should be conserved at any given time.

And we see, in light blue, that the total is conserved, that despite the peaks and valleys of the other forces, the sum is relatively constant. The slight error could be accounted for by inaccurate spring constant measurement, that the height isn't exactly calibrated with the motion sensor, and the spring itself isn't perfect (it could have flopped during the experiment). But in all honesty, this is as good as we should expect.

Lab 11: Work-Kinetic Energy Theorem

According to theory, the work done to move an object turns into kinetic energy, therefore, if we limit the amount of air resistance and friction, we should be able to find a direct identity relationship between work and kinetic energy.

To do this, we need a motion and force sensor, since kinetic energy is a function of velocity, and work is a function of force. We deduce the kinetic energy after capturing velocity and figuring out the mass of the system, applying the formula for it. We find work by noting the integral of force and, by plotting both against the distance, we could compare work and kinetic to see if they are indeed the same, such that:

W = F * d = (m v²) / 2

The force sensor is clamped to the far end of the picture above, connected to a spring and a weighted cart over a track. The motion sensor is on the near end. First, we measure and zero the sensors with the spring uncompressed. Then, we pull the cart, such as to extend the spring an arbitrary amount--this will be recorded as positive distance. Thus, the chronological order of events will appear in reverse when the graph is plotted against position. When released, the stretched spring should be at maximum force, reaching zero at uncompressed (neutral) position,while velocity increases. Work is converted to kinetic energy.

This is our data:

The data points of negative position are crossed out, telling Logger Pro to ignore what happens after the spring reaches uncompressed position, and begins to be compressed. We create a calculated column for kinetic energy. And we use Logger Pro to find the area under the force curve between certain limits, and analyze the value of kinetic energy at that point. To illustrate, here are a few points. Note the similarity between the values:

Please excuse the moire, the resolution of the phone doesn't capture computer screens well. More importantly, note that work and kinetic energy as reported by Logger Pro diverges as time goes by, revealing the faults in our experiment. First, our spring isn't ideal, and hangs to one side off the track, even at rest, which could distort the zeroing of our sensors. Second, there are other forces at work in reality, such as friction. Our measurement time is brief, exacerbating error. Nevertheless, we see clear signs of a correlation, so we consider that the relationship between work and kinetic energy confirmed.