By:
Dan Bobkoff and Kelly McNamee

Collisions are a fact of life. People walk into others on the street. Cars crash into each other. Molecules collide. Every collision is physics in action. In our class, we created controlled collisions to measure the effects of motion and energy in such cases. We crashed rolling carts with students on them into each other. Standardized, emptied soda cans mounted on the front of the vehicles absorbed the energy.

In completing this exercise, we found that the energy required to deform the cans is directly proportional to the level of deformation squared.

Momentum is conserved in collisions, whereas some energy is
always lost. This is the expected and found conclusion of a dayÕs worth of
collision and days of analysis. Momentum is the speed (velocity) of an object
multiplied by the mass and taking into account direction. The formula for
energy is (0.5)mv^{2}.
This formula is useful in determining the loss of energy in such
collisions.

The experiment commenced on the University of Rochester quad where we set up two identical carts with four identical empty soda cans securely mounted on the front of each. The carts are positioned at opposite ends of a straight track with the cans opposing each other. A member of the class sat on each cart, and a member was assigned to push the students as necessary. Each student was weighed and each can was measured before and after the deformation.

We carried out a variety of collision situations. Some sent the carts towards each other at varying rates. Others left one cart at rest, while the other was pushed into it. Each collision was videotaped in the same manner for later analysis.

Analysis was completed over several days. Using special
software called VideoPoint, we established the velocities before and after the
collisions for each cart. The software works by having the user tell the
computer a fixed distance on the image to use as a reference. Then, for each
frame of the video, the experimenter tells the computer the amount the cart
moved. This is done by clicking on a part of the cart that moves. The same part
of the cart must be used to get accurate results. For example, many clicked on
the front wheels. The computer then produced tables and graphs from the result.
The process is imperfect because of the limitations of the video and computer
systems, and natural human error. From this data, we used formulas and
extrapolation to find momentum and energy and analyzed using spreadsheet
software.

While VideoPoint is inexact, it does provide a multitude of information. It gives velocity data for every frame and every cart. The amount of data expands quickly. We, however, wished to look at only a small portion of the dataÑthat being the information immediately before the collision between the two carts, and that immediately after the collision.

VideoPoint gave us a velocity graph for each of the carts. We pinpointed the three points before and after the collision and found an average for each. Two example graphs from Run #3 are shown below.

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In most of the runs, more than one person analyzed the velocity information. Therefore, we had to find averages and error readings for velocity-initial of the right cart and velocity-initial of the left cart, as well as the final velocity of the left and right carts. We did this by first finding the averages and then, to find the error, we subtracted the smallest velocity of that category from the largest velocity and divided this result by 2.

Then we used these averages to find the initial momentum and
final momentum of each run. The errors of each velocity were important when we
had to find our momentum error calculations. First, the way we found momenta
was to multiply mass by velocity. Since we had two different velocities and
masses to work with when we were finding each momentum reading, we took mass_{1
}multiplied by velocity_{1} and added this to mass_{2}
multiplied by velocity_{2}. For example, in Run #3, our first run
together, we found initial momentum by multiplying the initial velocity of the
right cart (-1.524) by the mass of the right student (135) (note that the carts
cancel each other out so their rate is moot) and added this result to the
initial of the left cart (2.528) multiplied by the mass of the left student
(174) to bring us to the initial momentum (234.113). The final momentum is
calculated in the same manner using the final velocity in place of the initial.
We then found the error of the momentum by finding out what the maximum
momentum would be when we take into account the error in the velocity measures.
This was subtracted from the minimum of the velocity measures. The answer was
divided by 2 lending us our error measurement for momentum. We did this by
finding momentum with velocities plus their errors, and then subtracting this
by the result of finding momentum with velocities minus their errors. Then, we
took this result and divided by 2 to find our momentum errors.

Because of the high level of error in our velocity results, and therefore our momentum results, it is difficult to state whether momentum is conserved or not. When forced to give an answer the best estimate we can provide is that momentum is conserved. We believe this is true because we see no reason for momentum not to be conserved and it is possible to say this within our error estimates.

Next, we found the initial kinetic energy, the final kinetic energy, and the loss of kinetic energy for each collision and the corresponding error reading for each. First we found the initial kinetic energy with this formula:

((0.5)mass_{left}*velocity_{initial-left}^{2})
+((0.5)mass_{right}*velocity_{initial-right}^{2})

Final energy uses the same formula, except that it substitutes final velocities for the initial shown in this formula. The kinetic energy lost is found by taking the final kinetic energy and subtracting the initial final kinetic energy.

We found the error of the initial and final kinetic energy the same way. We used the same energy equation, but substituted the error of the velocity that corresponds to the velocity in the above formula:

((0.5)mass_{left}*error of velocity_{initial-left}^{2})
+((0.5)mass_{right}*error of velocity_{initial-right}^{2})

We found the error of the kinetic energy lost by taking the maximum of what the error lost would be when taking into account the errors in the finals and initial kinetic energy. Then, we subtracted the minimum. This result was divided by 2 lending us our error:

MAX = (E_{final} + error) Ð (E_{initial }Ð
error)

MIN
= (E_{final} Ð
error) Ð (E_{initial }+ error)

Error of Energy Lost =

(MAX Ð MIN) / 2

Then, we found out
how much each can deformed in each collision. This was done by taking the
standardized 10.4cm length measurement and subtracting the final measurements
of the length of each can. We then added these measurements together to find
the totals for each collision. This total was then squared. We found this total
and squared it to try and find which of the two models we were given explained
the relationship between the deformation of the cans and the loss of kinetic
energy. One model was that the deformation of the cans had a linear
relationship to the loss in kinetic energy. The second model was that the
deformation of the can squared had a linear relationship to the loss of kinetic
energy. To establish this, we graphed both the deformation versus the loss of
kinetic energy and the total deformation squared versus the loss of kinetic
energy. The charts are shown below. _{}

Although neither is a true linear relationship, the
deformation^{2 }graph provides a more consistently measurable line that
goes through the origin. Three points seem like outliers. These three points
are the ones to the bottom left of the graph. They are inconsistent with the rest of the points because,
as deformation increases, loss of kinetic energy should also increase. These
points do not show this to be true. We could find no consistent error in
calculations to explain this anomaly. The only reason we can attempt is that
the more cans that were deformed in each collision, the better the data seems
to be. These three runs, 1,3, and 9, had fewer deformed cans than other runs.

Accompanying this lab report is the spreadsheet containing all of our data.

**Conclusion**

As well as our equipment and experience can conclude,
momentum does seem to be conserved in violent collisions between two objects.
The loss of energy seems to be proportional to the deformation^{2}. A
more accurate video and computer system would help further prove this finding,
but with the large amount of data we worked with and the careful error
calculations, we are fairly confident in standing by that result.

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**Remarks**

This experiment was the culmination of weeks of hands-on physics education. This activity allowed us to be a real part of the experiment (quite literally) and then get practice analyzing large amounts of data in a manner that built on our past experiences in the class.

The class progressed nicely from basic experiments to more
complex analysis. We both feel that our analytic scientific abilities have
increased immensely. In addition to that, we learned and reviewed valuable
computer and math skills. This experiment and the class have, on the whole,
been a success. _{}