Frank Mazzarella and Jacqueline Boyce

Final Lab Report

May 5, 2003

**Abstract**

**-**The object of this
experiment was to learn more about collisions. By using humans on carts, we gained a more hands on
experience with collisions and the analysis of them.

**Theory**

**-**There are many
theories behind this experiment, all pertaining to collision, most importantly
about the conservation of momentum and energy.

**-**In carrying out
this experiment about collisions, we follow several given physics
formulas. Firstly, we looked at
velocity, and we know that **v=distance*time**. We also know that the
formula for error associated with velocity is:** error= (max-min)/2.** Next
we looked at momentum. The formula
for momentum is: **P=m _{1}v_{1}+m_{2}v_{2 }**and the formula for error is:

-These are all of the pertinent formulas we used in our experiment,

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**Purpose**

-The purpose of this experiment was to explore and gain an understanding of the conservation of momentum, the conservation of kinetic energy and the loss of kinetic energy using human collisions while on carts. Also, to determine a relationship between the loss of kinetic energy and the deformation of the cans attached to the front of the carts. After completing the physical aspect of the collisions we then used film from video analysis to collect data.

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**Experiment**

-In order to complete this experiment we had to first determine a series of factors. First we measured the dimensions of the cart, as we needed to know the distance between the two sets of wheels for our video analysis. This distance is recorded at 81.5 cm. Then we collected the masses of all students involved in the collisions, which are recorded on Chart 1. Then we measured the length of the cans that we would attach to the front of the carts which were 10.4 cm. We would later use the deformation of these cans to collect data about the collisions. Now we were ready to begin the actual collisions. We completed 18 collisions using different initial conditions. Two of these collisions were deleted because of improper procedure (Runs 8 & 13) for example, Frank jumping off the cart. Once each collision was completed we measured and recorded the deformation of each can per cart which resulted in a total of 8 separate measurements per collision. These measurements are shown in Chart 1. From here Professor Wolfs digitized the videos of the collisions and put them on cd-rom. This allowed us to analyze this video through video analysis techniques. For each collision, at least two people involved in the collision analyzed the video and determined a best estimate value for the initial and final velocities of each cart. From here we took the data and determined the error in the velocities, the momenta before and after each collision and the error associated with these momenta, the total kinetic before and after each collision and the error associated with this, and finally the loss of kinetic energy in each collision and the error associated with this loss. Then we attempted to find a relationship between the deformation of the cans and the loss of kinetic energy. In doing so we furthered our knowledge of the conservation of momentum, the conservation of energy, and many other factors involved in collisions.

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**Data Analysis**

-All of our data is compiled in Chart 1 which is
attached. We determined the best
estimates for the velocities of both carts before and after the collision, the
error associated with these velocities, the momentum before and after along
with the error associated with these momenta, the total kinetic energy before
and after, and the error associated with these values as well, and the loss of
kinetic energy and the error associated with this loss. In order to find the best estimate for
the velocity before, we took the average value of the velocities before
collected by person one and person two.
We did this for both carts and followed the same procedure to find the
best estimate of the velocity after.
To determine the error in the velocities, we used the error formula that
**error= (max-min)/2**

-Next we found the momenta before and the momenta after the
collision. We determined this
momentum by using the formula **P=m _{1}v_{1}+m_{2}v_{2}**

We determined the error associated with the momenta before and after for each collision. These values are shown in Chart 1.

-We then went on to determine the total kinetic energy before and after the collisions. We determined this by using the formula for total kinetic energy which is;

**Kinetic energy=1/2m _{1}v_{1}^{2 }+
1/2m_{2}v_{2}^{2}. **We then determined the
error associated with the kinetic energy before and after, using the formula

-Finally we determined the loss of kinetic energy. The formula we used was **loss of
kinetic energy=total kinetic energy before-total kinetic energy after**. We
determined this for each collision and then determined the error associated
with this loss. This error value
was determined by adding the errors associated with the kinetic energy before
and after the collision for each cart.

*In greater detail, we will now discuss Collision #1 because both of us were involved.

-Collision #1 consisted of the left cart with a mass of 140
lbs. heading towards the other person with a velocity of 2.75 m/s^{2}. The cart on the right with a mass of
132 lbs. was heading towards the left cart with a velocity of -1.85 m/s^{2 }. This value is negative because of the
direction in which the cart was traveling. After the collision the velocity of the left cart is .45 m/s^{2
}, and the velocity of the right cart is -.05m/s^{2 } .
The momentum before the collision of this system was 140.8 g*m/s^{2 } with an error of 54.8. The momentum after this collision was 56.4 g*m/s^{2}
with an error of 79.6. Within our
error limitations momentum is conserved in this collision. The total kinetic energy before this
collision was 755.26 g*m^{2}/s^{2}. The total kinetic energy after this collision was 14.34 g*m^{2}/s^{2}. This resulted in a loss of kinetic
energy of 740.92 g*m^{2}/s^{2}. The error associated with this loss in kinetic energy is
equal to 59.14 g*m^{2}/s^{2}.

-The last thing that we needed to determine was a
relationship between the deformation of the cans and the loss of kinetic
energy. The two relationships we tested were loss of kinetic energy versus both
the deformation of the cans and the deformation of the cans squared. We calculated the total deformation and
the total deformation squared. We then
graphed both of these relationships in Microsoft Excel which are attached as
Graph #1 & Graph #2. According
to these relationships we decided that the loss of kinetic energy is
proportional to the deformation squared, as there is a clear and better
relationship within the error limitations. We determined that there was less of a correlation between
loss of energy and deformation as can be seen in the graph.^{ }

-As an example of how video analysis works, we also have included a graph (Graph #3) of speed versus time for collision #17 which we got from the video analysis program. This collision did not involve Frank, just Jacqueline. It is easy to see where the collision occurred. It is around 2.5 seconds, and there is a significant drop in speed for both carts at this time. This graph is included to help illustrate another method of data collection and analysis.

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**Conclusion**

*In conclusion this lab report has furthered our knowledge of collisions, momentum, and loss of energy. We also learned a significant amount about the determination of errors and what these errors imply. We determined that the loss of kinetic energy is proportional to the square of the deformation of the cans. We also answered all of the questions asked of us in the lab handouts.

*After we concluded writing this lab report we found a possible significant error. This error was in terms of the units used while writing this lab report. We speculate that several other members in the class fell into the same predicament because all throughout the creation and data collection of this report, nobody in the class caught notice of this error. We measured our masses in lbs. however; all other data collection was measured in terms of metric values. Therefore, there may be a problem with the way in which these numbers were converted, or lack of conversion. This may have offset several numbers within the data, but we are not sure. We assumed that since it was not mentioned in class and we were told to measure ourselves in lbs instead of grams the affect must not be too significant. We would be interested to see if the results were different however.

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**Remarks**

-We would like to thank you for the semester. Class was interesting and fun at the same time. Both of us are non-physics majors enjoyed taking this class.