Energy and Momentum
of Collisions
Relating
Momentum and Kinetic Energy to Can Deformation
Trip and Zach
Passenger Cart Collisions
(4/14/03-4/30/03)
Abstract
The
purpose of the passenger cart collisions was to use video imaging to analyze
inelastic collisions. This enabled us to develop or refute any relationship
between conservation of momentum and kinetic energy and can deformation from
the collisions of the carts. Our main results indicated that there definitely
was a relationship between can deformation and loss of kinetic energy in the
collisions.
Theory
The
underlying principals behind the actual collisions that took place existed in
the theories of conservation of linear momentum and kinetic energy and also in
can deformation magnitude. As a general assumption about the deformation of the
cans attached to the carts, we operated with two rules in mind when analyzing
our data: 1) the energy required to deform the can(s) is proportional to the
deformation and 2) that energy is also proportional to the square of the deformation.
The
theory for conservation of linear momentum in relation to our two passenger
carts is as follows:
m_{1}v_{1} + m_{2}v_{2
}= m_{1}v_{1}Õ + m_{2}v_{2}Õ
or
(m_{1}v_{1} + m_{2}v_{2})
Ð (m_{1}v_{1}Õ + m_{2}v_{2}Õ) = 0
The theory
states that the product of an objectÕs initial mass and velocity added to that
of the second objectÕs initial state should equal the sum of the final values
(or Òafter-collisionÓ state) of both objects.
With
respect to kinetic energy release, the following formula depicts how that value
is calculated:
E_{deformation} = Change
in K_{E}
Change in K_{E} = ABS[(½
m_{1}v_{1}^{2} + ½ m_{2}v_{2}^{2})
- (½ m_{1}v_{1}^{2Õ} + ½ m_{2}v_{2}^{2Õ})]
Thus, the
energy of deformation is equivalent to the change in kinetic energy from
initial kinetic to final kinetic states during the collisions.
Experiments
In
our experiments we used two students, two passenger collision carts with can
mounts, more than 150 cans for 18 trials (CAN FROM THE SAME MANUFACTURER), two
meter sticks, a scale, a camcorder, and VideoPoint video analysis software. The
experiment was setup up with a collision point (or reference point) picked at
an arbitrary location along a sidewalk. From this point, lines were drawn with
chalk the same distance away in opposite directions, and then one more set
another identical distance beyond that. This guide gave each cart pusher a
precise location of where to start pushing the cart and where to let go before
the cart and person coasted to the central collision spot. For each collision,
the passengerÕs mass was noted and the deformation of the cans was measured
can-by-can and then tabulated.
For
each collision, the video camera recorded the people in it, the trial number,
and the collision itself. This video was then spliced and converted to 18
separate movies (one for each run), which were then burned to a CD and analyzed
with VideoPoint software. During the analysis, a crucial screen calibration of
a known distance was made. This known distance was the measured length from the
front to back wheels of a cart. After calibration, the video could be broken
down into the two objects and their tracked movement between all frames. This
ball-park selection method provided the data for which the rest of the
experiment was constructed upon.
Data Analysis
Taking
Measurements
The
actual data plots taken from the video recordings were applied by hand in a
Òbest-guessÓ fashion based on our ability to follow the same pixel through all
frames. Of these data plots, the ones we used to form our ÒinitialÓ and ÒfinalÓ
calculations for velocity were located in sets of 4 or 5 points just before and
just after the collision, respectively, as illustrated in the following graph:
Figure 1: Run 2 (Trip Moving, Zach
Still) Velocity Plot
It
was easy to pick a collision point based on where the data points jumped up or
down to a new velocity value. From this location, one could project forward and
backward about five points to obtain an average value for both initial and
final velocity of the left and right carts.
Determining
Errors
Our
error determination came from comparing different measurements taken by each
person involved in their respective run. Four separate measurements were
analyzed for error: V_{i} left, V_{i} right, V_{f}
left, and V_{f} right. For example, the average values that I
calculated me for Run 2 were compared with ZachÕs average values and the Òplus
or minus errorÓ for that particular measurement was just half of the difference
between each set of measurements. The following table shows the velocity errors
in their entirety:
PLUS/MINUS ERROR |
||||
Run |
Vi L |
Vi R |
Vf L |
Vf R |
1 |
0.25 |
0.15 |
0.27 |
0.05 |
2 |
0.26 |
0.10 |
0.19 |
0.03 |
3 |
0.27 |
0.28 |
0.13 |
0.09 |
4 |
0.40 |
0.03 |
0.07 |
0.07 |
5 |
0.00 |
0.25 |
0.01 |
0.13 |
6 |
0.38 |
0.21 |
0.36 |
0.26 |
7 |
0.00 |
0.48 |
0.14 |
0.12 |
8 |
0.00 |
0.00 |
0.00 |
0.00 |
9 |
0.15 |
0.32 |
0.04 |
0.07 |
10 |
1.43 |
0.00 |
1.00 |
0.91 |
11 |
0.00 |
0.08 |
0.72 |
0.24 |
12 |
0.29 |
1.05 |
0.53 |
0.01 |
13 |
0.00 |
0.00 |
0.00 |
0.00 |
14 |
0.43 |
0.30 |
0.10 |
0.04 |
15 |
1.10 |
0.23 |
0.26 |
0.28 |
16 |
0.23 |
0.33 |
0.44 |
0.60 |
17 |
0.39 |
0.88 |
0.07 |
0.12 |
18 |
0.08 |
0.40 |
0.14 |
0.95 |
Figure 3: Velocity Error Calculations
As
Figure 3 shows, our errors were as extreme as ± 1.5 m/s and as small as ± 0.01
m/s. However, on average, our error was normally between
± 0.1 m/s and ±
0.5 m/s, which for all intensive purposes is acceptable for the level of
precision we could expect from the analog video camcorder we were using,
coupled with the manual data plotting in VideoPoint (Please see the external
Excel file for full access to the numbers used in the Figure 3 calculations). I
was important to note that Runs 8 and 13 were nullified due to a complete miss
in Run 8 and a crash in Run 13.
Examining
Conservation of Momentum
In
our experiment, it was necessary to use object mass (in our case it was just
the personÕs mass since both carts were identical in mass) and calculated
average velocity to produce average values for momentum. Excel formulas were
used to extrapolate a value of momentum (P) for each side (left and right) and
for each state (initial and final). The following spreadsheet sample provides
insight into how the momentum data was produced:
Figure 4: Average Momentum Value
Table
It
was clear from looking at P_{i} total and P_{f} total (the last
two columns) and then looking at Change in P (the last column) in Figure 4 that
conservation, although not proven with 100% accuracy here, was conserved. One
could assume that by looking at how large the values are for difference in
momentum that momentum might not be conserved. Below are the last three columns
in the Figure 4Õs data table:
RUNS |
Pi Total |
Pf Total |
Change in P |
1 |
156.06 |
134.27 |
21.79 |
2 |
610.53 |
658.74 |
48.21 |
3 |
234.11 |
178.10 |
56.02 |
4 |
35.33 |
24.77 |
10.55 |
5 |
-161.68 |
-194.08 |
32.40 |
6 |
-179.34 |
35.22 |
214.55 |
7 |
-398.64 |
-360.81 |
37.83 |
8 |
0.00 |
0.00 |
0.00 |
9 |
-89.27 |
-62.41 |
26.86 |
10 |
335.48 |
371.83 |
36.34 |
11 |
-56.13 |
-256.03 |
199.90 |
12 |
327.34 |
250.32 |
77.02 |
13 |
0.00 |
0.00 |
0.00 |
14 |
11.38 |
-21.48 |
32.86 |
15 |
147.10 |
138.28 |
8.82 |
16 |
-125.86 |
-197.67 |
71.81 |
17 |
12.91 |
-145.21 |
158.12 |
18 |
-76.57 |
-196.87 |
120.30 |
Figure 5: Average Momentum
Calculations
However,
a larger Change in P was justified by larger values of momentum in most cases,
so the differences were all of the same proportions across all 18 runs. There were
exceptions to the trend (see Runs 6, 11, 17, 18) and these values were most
likely obtained because of systematic error in the way that the VideoPoint data
were manually plotted by the respective participants.
Examining
Conservation of Kinetic Energy
We
expected that there would be a loss
of kinetic energy (kinetic energy is released in the inelastic collision
through the crushing of the cans and also through friction). The following data
table illustrates our kinetic energy calculations based on our average value
velocities used to obtain momentum:
Ki L |
Ki R |
Kf L |
Kf R |
Ki Total |
Kf Total |
Change in Ke |
547.82 |
210.23 |
13.58 |
19.97 |
758.05 |
33.55 |
-724.50 |
846.89 |
0.49 |
253.39 |
269.67 |
847.38 |
523.06 |
-324.32 |
555.78 |
156.67 |
16.35 |
39.04 |
712.45 |
55.39 |
-657.06 |
296.84 |
201.36 |
0.51 |
0.60 |
498.20 |
1.11 |
-497.09 |
0.00 |
181.52 |
62.03 |
41.45 |
181.52 |
103.48 |
-78.04 |
352.41 |
568.94 |
20.91 |
29.17 |
921.35 |
50.07 |
-871.27 |
0.00 |
601.95 |
142.51 |
51.11 |
601.95 |
193.62 |
-408.33 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
313.47 |
415.36 |
4.49 |
2.19 |
728.83 |
6.68 |
-722.15 |
274.51 |
0.00 |
134.90 |
71.82 |
274.51 |
206.72 |
-67.79 |
286.98 |
279.08 |
62.67 |
34.07 |
566.06 |
96.74 |
-469.33 |
775.33 |
191.51 |
81.24 |
15.75 |
966.84 |
96.99 |
-869.85 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
245.44 |
238.21 |
10.93 |
4.34 |
483.65 |
15.27 |
-468.38 |
125.22 |
2.60 |
10.13 |
17.67 |
127.82 |
27.80 |
-100.02 |
239.93 |
344.50 |
18.33 |
38.59 |
584.43 |
56.93 |
-527.50 |
293.28 |
202.33 |
14.46 |
20.00 |
495.61 |
34.46 |
-461.15 |
136.47 |
213.56 |
15.52 |
101.73 |
350.03 |
117.24 |
-232.78 |
Figure 6: Average Kinetic Energy
Values
Our
assumption that kinetic energy was NOT conserved was correct because ALL values
of the change in K_{E} were negative. It was evident that K_{i}
total was larger that K_{f} total in every case.
Analyzing
Error for Both Momentum and Kinetic Energy Change
In
both series of calculations, kinetic energy and momentum value for initial left
and right and final left and right were added to get initial and final totals
so that an overall value of change could be produced. The table below shows
minimum and maximum versions of these values; they were calculated from
subtracting and adding the plus/minus velocity errors to and from the original
average values to get two more tables of identical labels (See attached excel
file for complete calculations):
Figure 7: Min/Max-Generated Error for
K_{E} and P
The
error in K_{E} calculations was obtained by halving the absolute
difference of the min and max values; P error was reached in the same manner.
The K_{E} error ranged from about ± 50 J to ± 300 J. The error for P
was anywhere from ± 7 kg m/s to ± 150 kg m/s. All runs were performed in at
least slightly different conditions, and at most at completely different
speeds. This fact accounts for the varying magnitude and value of the errors between
the runs and makes it irrelevant to compare them beyond the fact that they are
all in proportion to their respective values of momentum and kinetic energy, as
was explained in Figure 5.
Relating Can
Deformation to Kinetic Energy Loss
All
can deformation values were measured in centimeters of defamation from a full
ÒnormalÓ length of 10.4 cm. The following table shows the individual can
defamation values and then the TOTAL deformation in bold:
Figure 8: Can Deformations
We
observed that, for example, in Run 1, where both objects were moving at each
other at similar opposite velocities, the total deformation of the cans was
similar, too, in value. However, in Run 2 and others alike where one objects is
moving and hits another stationary
object with zero velocity, the deformation of the cans is drastically
different. To us this suggested that there was a direct, linear relationship
between the loss of kinetic energy and the magnitude of can deformation.
We
started out with total can defamations from the left and right, so it only made
sense to add left and right defamations to obtain a grand total to compare with
Change in K_{E}. From our theory (See Page 1) we knew that defamation
was proportional to kinetic energy and kinetic energy squared. So in order to
create a graph that displayed a linear relationship, we would have to
experiment with the following mathematically-manipulated values of K_{E}:
Run |
LEFT
total |
RIGHT
total |
TOTAL
Def |
Ke |
Ke^2 |
1 |
15.2 |
14.1 |
29.3 |
-724.5 |
5.2E+05 |
2 |
8.4 |
15.4 |
23.8 |
-324.3 |
1.1E+05 |
3 |
3.5 |
2.1 |
5.6 |
-657.1 |
4.3E+05 |
4 |
15.4 |
17.0 |
32.4 |
-497.1 |
2.5E+05 |
5 |
9.6 |
9.5 |
19.1 |
-78.0 |
6.1E+03 |
6 |
22.0 |
25.6 |
47.6 |
-871.3 |
7.6E+05 |
7 |
2.2 |
22.9 |
25.1 |
-408.3 |
1.7E+05 |
8 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0E+00 |
9 |
11.8 |
15.5 |
27.3 |
-722.2 |
5.2E+05 |
10 |
10.6 |
9.5 |
20.1 |
-67.8 |
4.6E+03 |
11 |
26.2 |
24.0 |
50.2 |
-469.3 |
2.2E+05 |
12 |
26.7 |
31.8 |
58.5 |
-869.9 |
7.6E+05 |
13 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0E+00 |
14 |
25.1 |
25.7 |
50.8 |
-468.4 |
2.2E+05 |
15 |
16.4 |
11.3 |
27.7 |
-100.0 |
1.0E+04 |
16 |
26.7 |
26.1 |
52.8 |
-527.5 |
2.8E+05 |
17 |
17.5 |
17.3 |
34.8 |
-461.2 |
2.1E+05 |
18 |
30.7 |
4.8 |
35.5 |
-232.8 |
5.4E+04 |
Figure 9: Comparing Total Deformation
to K_{E} and K_{E}^{2}
Once
we had the K_{E} and K_{E}^{2} values tabulated with
total deformation, we could set up a visual representation to see if indeed
there was or was not a linear relationship in our data:
Figure 10: Def. Total/K_{E}
Graphical Analysis
The
graphical representation of the data when looking at total deformation versus K_{E}
indicates a probable linear but indirect relationship. As total deformation of the cans increases,
the amount of kinetic energy gets increasingly negative (more is released).
When plotted against K_{E}^{2}, however, there is a linear but direct relationship. These graphs, although
there are some extreme outlier data points, do illustrate a pattern that
suggests a linear relationship.
Conclusion
We
had several underlying theories governing the behavior of the carts. The first
was the conservation of momentum. Our data suggested without reasonable double
that momentum was conserved. The second theory, conservation of kinetic energy,
was disproved. As expected, in all runs kinetic energy was released (negative)
because there was defamation in at least 1 can every time (which was obvious
from the inelastic behavior). The last theory, that of the linear relationship
between total deformation and kinetic energy, was an indirect one based on the
sign value for kinetic energy decreasing (getting increasingly negative) as
deformation went up. That was also expected and backed up the assumption that
the more the cans were crushed, the more kinetic energy had to have been
released.
Remarks
The
experiment itself was a fascinating conclusion to the weeks of work did on
collisions and the conservation laws. I was impressed with how simple the
actual trials were compared to the wealth of data and variables that lay
beneath the collisions. This lab offered not just a recapitulation of the
concepts we already were familiar with, but made us think about what those
principals meant abstractly.