Conservation of momentum for elastic
collisions in an isolated system
G.K Cheung
Department of Physics and Astronomy
University College London
21 December 2014
Abstract
The conservation of momentum and energy has been investigated by calculating momentum and
measuring angles obtained from colliding two spheres on a custom made rail bracket in an almost
isolated system. The two separate calculations of momentum m1v0 = (0.0176±0.0020) kgms-1 and m1v0
= (0.0173±0.0020) kgms-1 are almost inside of the experimental uncertainty, thus suggesting a possible
source of systematic error. Two out of eight measurements of the angle φ are outside of the
experimental uncertainty of the accepted value of φ = 90°, which is the angle required to confirm the
conservation of energy, suggesting another possible source of experimental error.
Introduction
The aim of this experiment is to observe the collision between two ball bearings, to investigate the
conservation of momentum and energy in an almost isolated system (sum of external forces is equal to
zero). The experiment is carried out on a custom made rail track with a pedestal and adjustable tracks,
one of the balls (m1) is released from a set point on the track, and the other (m2) is placed on the
pedestal. The balls will collide and the landing points of each ball bearing are recorded on a sheet
recording paper, which is placed under a piece of carbon paper. As the ball rolls down the track, the
ball gains speed, thus gaining momentum. As seen from:
p = mv
(1)
Where the momentum, p, is determined by the product of the mass and velocity of the ball bearing
[1]. Assuming that the momentum is conserved for this collision,
m1 𝑣⃗0 = 𝑚1 𝑣⃗1 + 𝑚2 𝑣⃗2
(2)
This can also be expressed in parallel and perpendicular components to 𝑣⃗0,
m1 𝑣0 = 𝑚1 𝑣1 cos 𝛼 + 𝑚2 𝑣2 cos 𝛽
(3)
The initial momentum of m1 before the collision must be equal to the total momentum of the two ball
bearings after the collision . Where v0 is the velocity of a ball bearing released from the set point
without any collisions. This means that the conservation of momentum could be determined by
measuring each of the different velocities (v0, v1 and v2), and using equation (2) to confirm if momentum
is conserved.
To investigate the conservation of energy, consider the masses of the ball bearings are equal.
Equation (2) will then reduce to:
𝑣⃗0 = 𝑣⃗1 + 𝑣⃗2
(4)
The relationship between the magnitudes of the vectors in equation (4) is given by the cosine rule[1]:
v02 = 𝑣12 + 𝑣22 − 2𝑣1 𝑣2 cos 𝜑
Landing
v2
Point 0
(5)
point 1
v0
α
β
φ
Landing
point 2
v1
Figure 1: The relationship between velocities of equal masses after an elastic collision
Now consider the collision to be perfectly elastic, the kinetic energy will be conserved [3],
1
2
𝑚𝑣02 =
1
2
1
𝑚𝑣12 + 𝑚𝑣22
(6)
2
Therefore, it will reduce to,
v02 = v12 + v22
(7)
Equation (7) turns out to be the same as equation 5 when φ = 90°.
This suggests that energy will be conserved when φ = 90°, hence allowing the investigation of the
conservation of energy simply by measuring φ, and see if it equals to 90°.[4]
Method
The schematic diagram of the system in the experiment is shown in figure 2, it is a custom made rail
bracket, with a track that could be adjusted using blue tack.
Point 0 is determined by inserting an
optical pin in the hole of the pedestal, to determine the initial point of collision. The height of the
pedestal is then adjusted to ensure meter on the track.
d0 =20.9±0.05cm
v0
m1
m2
h1=8.1±0.05cm
Landing points
h2= 19.1±0.05cm
Pedestal
Point 0
Recording sheet &
Carbon Paper
Figure 2: The schematic diagram of the conservation of momentum system
m1 and m2 are measured using an electronic scale, which are both m1 and m2 = (16.3±0.05)g.
Releasing the masses in this experiment are all done with a ruler instead of using hands to ensure
reliability. m1 is initially released from h1 = (8.1±0.05)cm above the horizontal plane of collision,
without anything on the pedestal. M1 will fall with a horizontal velocity on to the recording sheet. This
1
gives the distance between point 0 and v0, d0 = (20.9±0.05)cm. Using h2 = 𝑔𝑡 2 , where g is the
2
gravitational acceleration on Earth, the time of flight could be found, thus v0 could be obtained using
v=
𝑑
(8)
𝑡
The experiment could then be done with both m1 and m2, where the previous steps apply but with
m2 on the pedestal. Releasing m1 and colliding with m2 gives two landing points on the recording
sheet, assign the pair of results with a number. The distances between the landing points and point 0 are
then measured, giving values of d1 and d2. Repeat the experiment for 5 landings as opposed to the 10-20
landings from the established procedures for each angle; adjust the angle 5 times, to ensure clear results.
Multiple values of v1 and v2 could then be obtained using equation (8) where t is the same as the one
in the calculation of v0.
The conservation of momentum could be determined by comparing the values of m1v0 from equation
(2) and the value of m1v0 obtained from multiplying v0 from equation (8) with m1.
Conservation of energy could be determined through the measurement of φ as seen in fig.1, and see
if the measurement matches with φ = 90°.
Results & Analysis
The velocities of the two spheres after collision is shown in table 1. It is calculated using equation (8)
∆𝐴
∆𝐵
𝐴
𝐵
and the uncertainty is propagated using ∆z = z√( )2 + ( )2 .
-1
d1/m
v1/ms
d2/m
v2/ms-1
0.132±0.001
0.670±0.090
0.065±0.001
0.330±0.040
0.118±0.001
0.599±0.080
0.109±0.001
0.553±0.070
0.085±0.001
0.431±0.060
0.134±0.001
0.680±0.090
0.064±0.001
0.325±0.040
0.142±0.001
0.721±0.100
Table 1: Speed and distances from point 0 to the landing points
The velocities could then be used to calculate the momentum for each of the spheres after the
collision, using equation (1), hence allowing the calculating of the total momentum before the collision
using equation (2). The uncertainty of the momentum is propagated using ∆z = √(∆𝐴)2 + (∆𝐵)2 . The
momentums of the spheres are shown in table 2.
m1/kg
m2/kg
m1v1/kgms-1
m2v2/kgms-1
m1v0/kgms-1
0.0163±0.0001
0.0163±0.0001
0.0109±0.0010
0.0054±0.0010
0.0163±0.0020
0.0163±0.0001
0.0163±0.0001
0.0098±0.0010
0.0090±0.0010
0.0188±0.0020
0.0163±0.0001
0.0163±0.0001
0.0070±0.0010
0.0111±0.0010
0.0181±0.0020
0.0163±0.0001
0.0163±0.0001
0.0053±0.0010
0.0118±0.0020
0.0171±0.0020
Table 2: Mass and momentum for each sphere after the collision and momentum before collision
The weighted mean of m1v0 gives m1v0 = (0.0176±0.0020) kgms-1, which when compared with the
results obtained from multiplying v0 with m1, m1v0 = (0.0173±0.0020) kgms-1, they roughly agree with
one another within one standard deviation. The slight difference between the values suggests a
systematic error in the experiment. The suspected source of such error could be the spheres hitting the
pedestal during their fall. This causes some of the energy to be transferred into the pedestal and also an
upwards projection is caused by the reaction force on the pedestal.
By comparing and seeing if the angles of φ = 90°, the conservation of energy could be concluded.
The angle φ for each pair of points is measured using a protractor, and the anomalies are excluded. The
measured angles for each pair of results obtained would be displayed in table 3.
Angles/°
90±0.5
131±0.5
93±0.5
131±0.5
130±0.5
90±0.5
140±0.5
89±0.5
Table 3: The angles 𝛗 measured from the experimental results
Only two out of eight results measured equals to 90°, this overall does not agree with the expected
value ofφ = 90°. This suggests a systematic error in which the suspected source of error could once
again be the spheres hitting the pedestal. The large angles only occur when it is far away from point 0,
but all the points nearer to the point 0 are at 90°or somewhat close to 90°. This suggests that because
some energy is transferred into the pedestal and an upwards projection is caused upon hitting the
pedestal, the spheres further away from point 0 could not travel as far as it theoretically could. This
explains the large angle which otherwise be close to 90° if it had travelled its full length.
Conclusion
In conclusion, the data suggests that the momentum is conserved, as m1v0 calculated using v0 roughly
agrees with the m1v0 calculated with equation (2) within one standard deviation. Although it does not
fully agree, but it is suspected to be a systematic error caused by the spheres hitting the pedestal during
their fall. The data also suggests that energy is not conserved and the collision was not elastic because
only two out of eight results agrees with the expected value ofφ = 90°. Anomalous results also occur
when the track angle is not very wide, which suggests that having a small angle allows the sphere to hit
the pedestal hence anomalous results are produced. Both of these suggest that a systematic error is not
accounted for. From the results we could predict that any alterations in the experiment such as
changing the release height would not cause any major differences in the final results, and that the
momentum could still be seen as conserved. And that energy might be conserved if the pedestal
problem was fixed through proper adjustments to make sure the spheres do not hit it when falling. If
the energy is deemed not conserved even after fixing the pedestal problem, it can be said that the
energy is not conserved in an almost isolated system due to the loss of energy to the surroundings. The
experiment could be improved by increasing the number of repeats and using more precise measuring
instruments to increase reliability. The experiment could also be improved to have the pedestal already
fixed in place in a position where the spheres could not hit it during the fall.
References
[1] R.P. Feynman, R.B. Leighton, S. Mathew, The Feynman lectures on physics, Volume 1,
Addison-Wesley, 1964
[2] G.A. Wentworth, Plane and Spherical Trigonometry, BiblioBazaar, 2009.
[3] R.A. Serway & J.W Jewett, Physics for Scientists and Engineers, Brooks/Cole, 2004
[4] P.A. Barlett & J.R.Grozier, NX1 Conservation of Momentum, 2014