Conservation of Momentum and Energy in a Linear 1-Dimensional System
V. Ptakh – C. Pereira
velocities
Abstract
This paper aims to empirically determine
whether momentum and/or kinetic energy are
conserved throughout an entire 1-Dimensional
(1D) linear collision, both the elastic and
inelastic types. We demonstrate that for an
elastic 1D linear collision, modeled using an airtrack and two carts, momentum and kinetic
energy are both conserved. For an inelastic 1D
linear collision, employing the same model,
momentum is conserved but kinetic energy is
not.
and
, respectively.
Furthermore, their action/reaction pair of
forces will be
and
. Thus we have
the following set of equations:
Going back to our momentum, if we
differentiate the equation, we obtain the
formula for force:
1 – Theory
Theoretically, for an elastic collision, both
momentum and kinetic energy are conserved.
[1] Momentum for an object moving in a
straight line can be given by the following
equation:
Thus our set of equations now looks the
following:
(1)
Newton’s second law states that “an object of
certain mass m will undergo acceleration
equal to
, where
is equal to the
vector sum of all the individual forces acting on
the object.”[2] Combined with Newton’s first
law, it is safe to assume that in a collision a
certain force is exerted on one of the objects,
since its motion changes. Suppose two objects a
and b collide head-on in a perfect environment,
free of the burden of friction. The objects will
have initial velocities
and , and final
Now, using Newton’s third law, we can say that
where the negative sign
indicates direction. Now if we add the
momentum derivatives, we will find out that
the sum is zero – which directly indicates that
the total momentum has not changed. [3] The
general equation for the total momentum of an
elastic collision in perfect conditions is:
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Conservation of Momentum and Energy in a Linear 1-Dimensional System
(2)
V. Ptakh et al
2 – Experiment
The total kinetic energy of a colliding 1D
system1 can be written by the following integral
of the above formula:
(3)
For inelastic collisions, the general momentum
formula for collision is:
(4)
Let M =
. The equation for total kinetic
energy of the colliding inelastic system will be:
(5)
Rearranging for
, however, one can see that
the final total velocity calculated with the
energy equation does not equal to its
counterpart calculated with the momentum
equation. We know momentum is conserved;
therefore the offset of the energy value could
be blamed on no energy conservation. Thus,
for an inelastic collision.
1
It is important the system is 1D, since this formula would be
rendered useless in case of a change in height of the objects.
Fig. 1 The setup of the experiment. The right hand
cart was the one that was switched.
Before any reasonable experiment could be
conducted, one had to be convinced the track
was leveled. If such a procedure is not
conducted, there is a very high probability that
the experimental data will be wrong. This is due
to many reasons, the most obvious being the
fact that one end of the track will be at a
different height that the other, destroying any
notion of constant velocity, and thereby
rendering momentum calculations useless. The
other major problem would be higher airfriction. Although small, such friction still exists,
and it can reflect significantly on the data.
The track was leveled in the following manner.
One cart was placed on the air track. It was
released with some force applied, enabling it to
make a few traversals. Meanwhile, the Motion
Sensor was measuring the cart’s velocity. A
graph was plotted, of velocity versus time. A
section where the cart was moving towards the
sensor was taken, and the slope calculated.
Then, a segment where the cart was moving
away from the sensor was taken, and the slop
calculated. Ideally, the two values for the slope
should be equal. However, that would be rather
hard to accomplish so we had to be satisfied
with the fact that the values were equal within
error bounds.
After the track was leveled, the actual
experiment could be performed. The carts were
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Conservation of Momentum and Energy in a Linear 1-Dimensional System
placed on the air track, and given some initial
speed using brute manpower. If the air-track
was leveled well enough, the initial velocity
should stay almost the same throughout its
motion before colliding. We have used a variety
of experiments to simulate the collisions, in
order to be able to test almost all situations.
There were 8 experiments conducted in total. 4
of them were done with the carts having the
same mass, within error bounds. Furthermore,
out of those 4, 2 were elastic and 2 were
inelastic collisions. Finally, one of the two elastic
collisions was when one cart was standing still
and the other one when both were moving.
Similarly, one of the inelastic collisions was with
one cart still, the other when both were
moving. These 4 experiments are identical to
the rest of the 8, only the other 4 had their carts
at quite different masses. A tree diagram is
show in Figure 2.
g. 2 A tree diagram demonstrating the 8
Mass of carts equal
elastic
inelastic
Mass of carts
different
elastic
inelastic
one cart was still
both carts moving
one cart was still
both carts moving
one cart was still
both carts moving
one cart was still
both carts moving
Run #
#1
#2
#3
#4
#5
#7
#6
#8
V. Ptakh et al
3 – Data and Discussion
The following data has been obtained using
Data Studio and graph analysis.
Mean KE at different intervals + σ (Joules)
Mean of total P +
σ (kg ms-1)
before collision
1
-0.06 +/- 0.00
0.01 +/- 2.93e-4
during
0.000 +/5.74e-4
2
-0.01 +/- 0.01
0.041 +/- 0.002
0.005 +/0.002
0.001 +/- 2.376e-4
6.813e-4
+/1.895e-4
Run #
3
-0.0023 +/- 0.002
4
-0.016 +/- 0.003
0.003 +/- 4.397e-4
5.216e-4
+/2.684e-4
5
-0.046 +/- 0.003
0.005 +/- 3.574e-4
0.002 +/8.798e-4
6
-0.045 +/- 0.002
0.004 +/- 4.597e-4
0.003 +/8.302e-4
7
0.027 +/- 0.004
0.005 +/- 5.961e-4
0.002 +/0.001
8
0.010 +/- 0.004
0.002 +/- 2.816e-4
0.001 +/3.840e-4
after
0.01 +/3.26e-4
0.037
+/0.002
5.467e4 +/8.392e5
2.959e4 +/7.053e5
0.003
+/1.009e4
0.002
+/1.130e4
0.004
+/3.387e4
1.125e4 +/9.604e5
Fig. 3 Average of total momentum and average total
kinetic energy for specified intervals. Momentum
was taken to be conserved, with judgement based on
graphs obtained. Inelastic collisions are highlighted.
because the.
Fig. 2 A tree diagram demonstrating the 8
experiments performed. Note the order of run #7 and
#6. This occurred because the experiments were not
done in order. It would not matter usually, but
DataStudio, the program we used to plot the graphs
directly from the motion sensors, will not change the
numbering.
Fig. 4 An example curve showing Run #6.
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Conservation of Momentum and Energy in a Linear 1-Dimensional System
Figure 4 demonstrated the type of graph that
was obtained. Notice how a little earlier than 2
seconds, the topmost curve demonstrating
velocity significantly changes, and both carts are
moving at the same speed together. The middle
curve demonstrated momentum, and since this
is an inelastic collision, it can be clearly seen
that momentum did not change throughout the
collision. Finally, the lowest curve shows Kinetic
Energy, and one can clearly see that final kinetic
energy is much lower than initial. Thus our
theoretical postulate that
is proven to
be correct.
Notice that in Figure 3, for elastic collisions the
KE is conserved, with the values of KE before
and after collision either the same, or within
error bounds. It is also clear, however, that for
inelastic collisions, highlighted in gray, KE is in
no way conserved with final KE being very close
to 0.
4 – Conclusion
The experimental investigation of momentum
and energy conservation in a linear 1Dimensional colliding system has shown that
even in imperfect conditions, far from the
“ideal” conditions used for theoretical proofs,
momentum is conserved at all times. Similarly,
Kinetic Energy is also conserved but only for
elastic collisions. For inelastic collisions Kinetic
Energy is not conserved and the following
inequality holds:
It should be noted however, that despite the
conservation of KE in an elastic collision, KE is
conserved only before and after the collision.
During the collision, when energy is
transformed, KE is not conserved.
It can be seen in the data that for some data the
errors are as big as the data itself. This could
wither imply that the real value is zero, or
simply that there was a large deviation from the
average. Such things as Motion Sensor noise
V. Ptakh et al
can be blamed on this, giving zigzag patterns to
some of the momentum curves. In addition to
that, the air track may not have been leveled
adequately, yielding strange results. In the end,
however, it does not really matter, since in the
real world no collision is either perfectly elastic
or perfectly inelastic.
References
1. Randall D. Knight, Physics for Scientists
and Engineers: A Strategic Approach,
2nd edition, Pearson Addison Wesley,
2008 – p. 247
2. Ibid p. 146
3. Ibid p. 248