Analyzing Momentum Loss in Unbounded Systems

This experimental essay explores the concept of momentum loss in an unbounded system focusing on static friction. The study analyzes the relationship between mass, velocity, and frictional force by observing the change in momentum as toy cars collide with a stationary object. The findings highlight the influence of mass on momentum loss and demonstrate the importance of considering frictional forces when analyzing complex systems.

Ascertaining the limitations of the law of conservation of momentum
How does the change in mass affect the momentum loss in a toy car being ridden on
a wooden platform provided the temperature and applied force remains constant?
Subject: Physics
Word Count: 3744
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Abstract
This report hereinafter dwells on experimental results to determine the effect of
momentum loss given a change in weight of colliding objects. In this case, two toy
cars are used to determine the momentum losses. Normally, the focus has been
bounded on systems in which the net external force is often zero for momentum in
the system to be conserved. However, this report provides an experimental enquiry
into how the same would be experienced in an unbounded system where the net
force is no longer zero as the final momentum is less than the initial momentum
courtesy of frictional forces. The cars were allowed to move and collide on a wooden
platform. The obtained results were stunning. The momentum loss increased as
more weight was being added. This was due to frictional forces that were directly
proportional to the normal force. It should be noted that since the system was
horizontally oriented, and from Newton’s third law of motion, the normal force was
equal in magnitude to the effective weight of the toy cars. However, in future, in order
to obtain a more accurate finding, the parameters of the system must be redefined;
for instance, it is only kinetic friction which was considered whereas even static
friction has a huge bearing in the final total momentum of the system.
Word Count: 219
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Table of Contents
Abstract……………………………………………………………………………………………………………………………………..2
Introduction……………………………………………………………………………………………………………………………….4
Background Information…………………………………………………………………………………………………………….5
Research Question……………………………………………………………………………………………………………………..7
Initial Experimental Design ………………………………………………………………………………………………………..7
Modified Experimental Design …………………………………………………………………………………………………..8
Theoretical Hypothesis……………………………………………………………………………………………………………….9
Variables……………………….……………………………………………………………………………………………………………12
Experimental Procedure…………………………………………………………………………………………………………… 13
Data collection ……………..…………………………………………………………………………………………………………..15
Analysis of data …………………………………………………………………………..…………………………………………….18
Evaluation and Further Study ……………………………………………………………………………………………………18
Conclusion ……………………………………………………………………………………………………………………………….. 19
Works Cited……………………………………………………………………………………………………………………….… 21
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Introduction
Fundamentally, the law of conservation of momentum is premised on two postulates:
that mass is indestructible as well as energy hence it states that net change in
momentum is always zero, that is, momentum before collision is equal to momentum
after collision but only in an isolated system. However, more often, little regard is
given to the non-isolated cases. This paper will focus on the determination of the
effects of momentum changes with respect to external forces in a non-isolated
system. The investigation is done via an experiment. This paper will establish the
physical relationship between these two; the momentum loss and the mass change
in a non-isolated system.
Admittedly, in real world, the existing classical Physics concepts in this arena may be
inadequate hence need to develop further the concept of momentum loss in a non-
isolated system in the real world. For instance, frictional force which often
counteracts the applied force for the object to lose its momentum may have different
surface coefficients hence deviation from the practical scenario. Consequently, even
in a similar condition of temperature and force application, the motion of the object at
the points of contact between two surfaces may behave differently. The fact that
these forces are present in the real world and there being no relationship between
the two has incited me to look into how momentum changes when the normal force
of the objects changes hence, frictional force as well. This gives rise to the research
question “To what extent does changing the weight of an toy car (ranging from 0 to
100 grams in increments of 20 grams) going on a wooden surface affect the loss in
momentum (in kgms −1) with respect to the law of conservation of momentum, given
that the surface, temperature and force applied on the object remains the same?”
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This experiment will only take into account friction as methods to measure other
external forces are not within the scope of this research paper.
Background Information
The conservation of momentum is a fundamental concept of Physics. Momentum is
defined to be the mass of an object multiples by the velocity of the object and is
equivalent to the force required to bring the object to a stop in a unit length of time.
Sir Isaac Newton went further to prove that the rate of change in momentum is
directly proportional to the applied force and it takes place in the direction of the
applied force. For any array of several objects, the total momentum is the sum of the
individual momentum. This is the classical approach in determining the total
momentum for a system with many components. The conservation of momentum
states that, in an isolated system, the amount of momentum remains constant;
momentum is neither created nor destroyed, but can only change through the action
of forces as described by Newton’s laws of motion.
As pointed out earlier, conservation of momentum is premised on two distinct
universal postulates; that: we can never destroy mass and neither can we do the
same to energy. Notably, momentum is a vector quantity, having both a magnitude
and a direction just like force. In a complex system, analysis of momentum would
often be done by resolving the individual momentum using the static and dynamic
equilibrium equations. But this is only restricted to a bounded system so that
momentum would only be conserved in all three physical directions at the same time.
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Momentum loss springs from the systemic losses orchestrated mainly by the
frictional force factor. Now, friction is that force exerted by the surface of an object
when another object moves against it. Friction occurs in the direction opposite to
motion, and because of this, it is a force that affects the motion of objects. When a
box is pushed across a surface, friction works against the box in the direction
opposite to the box’s motion. This happens because when two objects rub together,
it sets off attractive forces between the molecules of the objects. For this reason,
friction can occur between any two objects. As friction results from attractive forces,
the amount of friction depends on the materials of those two interacting. For this
reason, all two surfaces have different coefficients of friction. Frictional force is
calculated by multiplying the coefficient of friction and the normal force in Newton.
However, for every two objects there are two coefficients of friction: static and
kinetic. Static friction is that which occurs when the object is at rest it prevents
motion instead of slowing it down. In all cases the force needed to overcome static
friction is much larger than the force needed to overcome kinetic friction (Cook, C.
L.). While kinetic friction is that which occurs when the object is moving and works to
reduce the motion.
Due to the fact the law of conservation of momentum only holds true when the
objects are in an isolated system, when friction is taken into consideration,
momentum is no longer conserved. This is because friction is an external force and
when it is present energy is lost due to the frictional forces which the object will
have to overcome and since the formula for momentum is the product of the
objects mass and velocity the momentum before will no longer equal the
momentum after.
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Research Question:
In this study, my intention was to undertake a scientific enquiry on momentum loss
for an unbounded system by considering the frictional factor hence the following was
the research question:
To what extent does changing the mass of a toy car (ranging from 0 to 100 grams in
increments of 20 grams) going on a wooden surface affect the loss in momentum (in
kgms −1) with respect to the law of conservation of momentum, given that the
surface, temperature and force applied on the object remains the same?
Initial Experimental Design
The experiment comprised of the following apparatus:
● Two toy cars with a large, flat body to add weights onto.
● Digital Balance in order to measure how much the toy cars weigh.
● Two Vernier light gates with cables in order to measure the velocity of both
the toy cars.
● Clamps in order to hold the light gates high.
● 10 masses of 20 grams to add to the cars.
● Wooden ramp of 2 meters for the toy cars to run on.
● Rubber bands to make a slingshot to pull the first toy car.
● Vernier Logger Pro with all cables to record the data
● Vernier Logger Pro Software to transfer the data to the computer.
● Marker in order to mark the start lines for the cars.
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Initially the experiment was to be conducted in a concrete platform. However,
coefficient of friction of concrete varied over a wide range and could increase the
margin of experimental error hence the choice for a wooden platform whose
coefficient of friction is known. The toy car is to be launched from a marked point by
rubber excitation and allowed to move to the next marked point; meanwhile, we
record the time taken to cross to the marked point and the distance covered is
allowed to be constant hence velocity can be determined; We assume that the car
moves at a uniform speed. Then we gradually increase its mass by 20 grams for
every 5 trials until the maximum mass is achieved.
Modified experimental Design
However, with this kind of velocity measurements, more errors are prone to recur
and hence contribute to systematic error that may affect the final outcome.
Therefore, two Vernier light gates with cables were used in measuring the velocity of
both toy cars hence improving the experimental accuracy.
Additionally, to restrict collision in a linear fashion, the two cars were restricted to
move in a straight path by providing guiding rails on their paths. This ensured that
only collision in one direction is considered as momentum is a vector quantity that is
designated by both magnitude and direction. This in turn simplified the experimental
approach that was taken.
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Theoretical Hypothesis:
It is hypothesized that as the mass on the car increases so is the force of friction.
The loss in momentum will be greater (Tiersten, Martin S.). Relevant equations
describing the physical relationship between these parameters will be provided
so as to get into the details. Notably, it is crucial to look at the equations
pertaining to this experiment.
Table 1: Normal Force
Mass of Toy Car A (grams [±0.01g]) Normal Force from Surface onto toy car
A (Newtons)
250.99 2.51
270.99 2.71
290.99 2.91
310.99 3.11
330.99 3.31
350.99 3.51
*Taking the acceleration due to gravity as 10 ms-2
Calculation of data
(i) Calculating the theoretical linear velocity and the theoretical
momentum loss:
In this case, we would like to calculate the linear velocities so that we make some
comparison with the one measured directly by the software hence equations of linear
motion:
V2= U2+2aS
But since the toy cars are originating from rest, then we have: V2= 0+2as
V= (2as)0.5(1)
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The acceleration a can be determined from Newton’s second law of motion: F= Ma
a= F/m
Now, F is the applied force to cause motion of the two cars to move towards each
other for collision, (this is to be determined from the experimental test results as
provided in table):
Consider the state of dynamic equilibrium of the car:
Hence Fr= Fd where Fr= Frictional force (considering kinetic friction) and Fd=
Applied force to cause the acceleration, but Fr = μFn
The coefficient of friction of wooden surface is given as: (0.5 +0.2)/2= 0.35
Then we determine the acceleration, for the 1st entry, a= F/m = 0.878/0.25099=
3.498m/s2
Now, referring to equation 1 above, we can determine the theoretical velocity
change:
V= (2as)0.5
Maximum S= 0.07 +0.11= 0.18m (S will be varied from 0.07 to maximum 0.18)
For example, the first entry, this is: v= (2x3.498x0.07)0.5= 0.7m/s
We now determine the momentum loss, is given by p= mv1-mv2= m(v1-v2) = mv’
Hence for the 1st entry mv’= 0.7x 0.25099= 0.1756 and these results are tabulated in
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table 2:
Table 2: Calculated Experimental parameter
Normal
Force (N)
Frictional
Force Fr
(N)
S
(m)
Drivin
g
Force
(N)
Acceleratio
n (ms-2)
Velocit
y (ms-
1)
Momentu
m loss
(kgms-1)
K.E
lost
(J)
2.51 0.88 0.07 0.88 3.50 0.70 0.18 0.06
2.71 0.95 0.09 0.95 3.50 0.79 0.21 0.09
2.91 1.02 0.11 1.02 3.50 0.87 0.25 0.11
3.11 1.09 0.13 1.09 3.50 0.95 0.30 0.14
3.31 1.16 0.15 1.16 3.50 1.02 0.34 0.17
3.51 1.23 0.17 1.23 3.50 1.09 0.38 0.21
(i) Calculating the kinetic energy lost
Linear kinetic energy is normally given as: 1/2Mv2
Hence we calculate for individual masses, for the first entry: ½x 0.25099x 0.72=
0.0615J.
As shown, it is hypothesized that the loss in momentum increases exponentially
as the normal force increases. This is so due to the fact that as the normal force
increases the energy lost to overcoming the frictional force is also increased
hence more speed is lost; therefore, the momentum lost increases.
A major issue with this however, is that it does not take into account the
coefficient of static friction, which is usually higher than coefficient of kinetic
friction and hence some energy would be lost overcoming the static friction
force. In addition to this, since the collisions are elastic, the sound the cars
would make would also reduce energy and as the momentum of the first car
increases, the sound it would make with the second car would also increase;
hence, higher loss in energy. Hence, the values would most likely shift
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downwards due to the coefficient of static friction and have a greater difference
due to the energy lost due to sound.
Variables
Dependent
Variable
Loss of momentum (kgms-1)
Independent
Variable
Mass (0 to 100 grams) (+-0.01g)
Fixed
Variables
Surface the car was on (Cardboard) in order to keep the friction
between the two surfaces the same. Using different surfaces is not
within the scope of this research paper.
Temperature by conducting the experiment in a controlled
environment with the AC at a constant temperature.
Force applied to the car before. This was kept the same for
consistency within each trial and to increase accuracy. It was kept
the same by making a slingshot with rubber bands that propelled
the first toy car.
Uncontrolled
Variables
Wear and tear on the tires of the car. This would cause the friction
to increase and hence cause greater inaccuracies within the
results.
As mentioned earlier, the independent variable was the weight of the toy cars which
was fortified by additional masses. Both the applied force and temperature were kept
constant; the former was controlled using a rubber band snap mechanism to excite
the movement of the cars simultaneously while the latter was made possible by the
fact that the temperature of the room was ambient. In order to minimize experimental
errors, the software was used to measure real-time velocity before and after impact.
Hence the only dependent variable was the momentum loss which was derived from
the difference between initial momentum and final momentum given the frictional
factor was at play.
Experimental procedures
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In order to gather the results, I first placed the wooden ramp on a flat surface so that
the normal force and force due to gravity cancel out completely. Then I setup the
data logger: I connected the in the two light gates into DISC1 & DISC2 respectively
and plugged in the logger pro to the power and used a USB cable to plug it into a
laptop. I started up the logger pro software, clicked ‘Open’ from ‘File’ and opened the
‘Photo gates’ package and then chose ‘Two gate timing’.
Figure1: Screenshot of Velocity measuring software
After this page opened up, I added the length of each of the cars 0.07m and 0.11m
for toy car A and B respectively. I made sure that the data collection was set on
time based and 18 seconds.
After this, I set up the light gates by using stands with clamps. I made sure that the
two light gates had enough space between them so that the toy cars had enough
space to collide; hence, I set them 33 centimeters apart as shown below. The light
gates were also placed 8.5 centimeters above the ramp.
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Figure 2: Experimental set up
Preliminary observations
As the experiment ensued, I noticed a well defined trend in the momentum loss as
the effective weight of the cars increased. Momentum loss became magnified with
increased weight. The sound being produced on collision also was magnified while
speed of the two cars was on a decline as the mass was being increased. Notably,
force being applied was kept constant such that the only independent variable
became the weight of the toy cars. After several trials, the results were obtained and
tabulated.
Data Collection
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Raw Data (2 d.p.)
Velocity of toy car A vs. Velocity of toy car B (ms-1)
Extra
Mass
Added
(grams
[+-
0.01g])
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average
0 1.43 0.25 1.47 0.28 1.48 0.29 1.46 0.34 1.43 0.25 1.45 0.28
20 1.22 0.22 1.15 0.13 1.38 0.20 1.26 0.25 1.14 0.27 1.23 0.21
40 0.98 0.19 1.11 0.16 0.95 0.16 1.00 0.21 1.05 0.23 1.01 0.19
60 1.10 0.20 1.14 0.20 0.92 0.05 0.96 0.16 0.88 0.12 1.00 0.15
80 0.81 0.18 0.53 0.20 0.73 0.15 0.71 0.12 0.77 0.11 0.71 0.15
100 0.68 0.08 0.61 0.09 0.71 0.19 0.63 0.15 0.64 0.13 0.65 0.13
Mass, Normal Force and Frictional Force of The Toy Tars
Extra
Mass
Added
(grams
[+-
Mass of
Toy Car
A
(grams
[+-
Normal
Force from
surface onto
toy car A
(Newtons)
Frictional
Force on
toy car A
(Newtons)
Mass of
Toy Car
B
(grams
[+-
Normal
Force from
surface onto
toy car B
(Newtons)
Frictional
Force on
toy car B
(Newtons)
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0.01g]) 0.01g]) 0.01g])
0 250.99 2.46 1.60 570.01 5.59 3.63
20 270.99 2.66 1.73 590.01 5.80 3.77
40 290.99 2.86 1.86 610.01 5.98 3.89
60 310.99 3.05 1.98 630.01 6.18 4.03
80 330.99 3.25 2.11 650.01 6.38 4.14
100 350.99 3.44 2.24 670.01 6.57 4.27
Average Speeds of different masses with Uncertainties (2 d.p.):
Average Velocity (ms-1) Uncertainty (ms-1)
Toy Car A Toy Car B Toy Car A Toy Car B
1.45 0.28 0.02 0.03
1.23 0.21 0.09 0.05
1.01 0.19 0.06 0.03
1.00 0.15 0.09 0.12
0.71 0.15 0.10 0.03
0.65 0.13 0.40 0.04
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Mass
of Toy
Car A
(grams
[±0.01
g])
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Toy Car A Uncertainty (ms-
1)
Toy Car B Uncertainty (ms-
1)
Figure 3: Graph of Uncertainty for Toy Car A and B
Table listing the momentum before, after and loss in momentum
Mass
Added
Momentum of Toy Car
A
Momentum of Toy Car
B
Momentum
Lost
0 0.364 0.160 0.204
20 0.333 0.124 0.209
40 0.323 0.116 0.207
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Graphical Representation
Mass
Added
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Momentum ofToy Car A
Momentum ofToy Car B
Momentum Lost
Figure 4: Momentum before, after and momentum lost
Data Analysis:
The graph shows a consistent increase in the loss in momentum as mass increases.
However, when the mass added equals 40, the trend is broken. This could be said to
be an anomaly in the data, however the standard deviation for the speed of the car is
only 0.03 ms−1 meaning that all the trials done for this mass were more or less
consistent; hence, there is no reasonable conclusion for this trend.
Evaluation and further study:
In order for the accuracy of this experiment to be greater, I believe that I would have
to calculate the center of mass of both the toy cars in order to place the weights
exactly on that area. This would result in no change in the center of mass hence, will
not contribute to more force being towards the end or towards the back. The fact that
I did not calculate the center of mass means that one set of wheels could have been
experiencing more friction than the other and that calls for great inaccuracy.
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Some systematic errors that were present were the use of a ruler to measure the
distances and width of the car. Compared to a digital measurer, a ruler has a much
higher uncertainty, hence making the experiment inaccurate.
Additionally, I feel that, even with the use of the slingshot to push the first car
forward, there could have been discrepancies with the force applied on the car. Even
the slightest difference could have caused inaccuracies and since rubber bands
were used, the tension could be so great that it caused deformation of the rubber
band. This could help explain the trend within the data collected.
Conclusion
In this experiment, where we have examined the effects of mass changes on the
momentum loss for an unbounded system, some fundamental conceptual
development has been set in motion. Initially, we have had to use mundane
analytical tools to describe momentum loss of an unbounded system. Hence this
extended essay has provided a basis onto which further work can be pursued. Most
importantly, the existential limitations with the current concepts made it impossible
for scientist to analyze complex system in an unbounded state. This work has
provided an impetus into discovering new and better methods to analyze complex
systems.
Consequently, design engineers can easily design these complex systems taking
into consideration both deterministic and probabilistic issues. Notably, further work is
needed to extend the study. As has been mentioned earlier, the scope of this
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experimental essay was limited to static friction, although kinetic friction also plays a
vital role in shaping the final impacts on the momentum loss. It was discovered that
as the mass changes, momentum of the system gets lost in a steady fashion.
However, due to experimental errors, the graph shows some anomaly in the general
assumed trend. This could possibly be as a result of the exaggerated experimental
methodology. For instance, we cannot ascertain with perfection the level of accuracy
of the modified means of measuring the velocity, although it is much better than the
earlier proposed means. Additionally, the platform onto which the toy cars were
ridden was assumed to have a uniform coefficient of friction; this may not be entirely
true. Notably, the net coefficient of friction results from the interaction of the car tires
with the wooden platform. Therefore, there should have been a means to first
measure the various coefficient of static and dynamic friction from which an average
value could be used for the purposes of calculating the frictional force.
Lastly, it was assumed that the normal force is entirely provided by the combined
weight of the toy cars (including the additional weights as experiment ensued).
However, this needed to be ascertained as from classical dynamics, during motion,
there are normally additional loads due to dynamic nature of the system as it moves.
Admittedly, however, from the experimental output, we can ostensibly restate that as
far as the scope and objectives of the experiment are concerned, there was a
significant effort to pursue the topic to its conclusion and that the findings obtained
here will offer a platform for further interrogation of the study in question.
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Worked Cited
"C.M. Energy And Momentum Versus Beam Momentum." Physics Letters B, vol 39, no. 1,
1972, p. 16. Elsevier BV, doi:10.1016/0370-2693(72)90847-7.
Cook, C. L. "Improved Version Of A Linear Momentum Conservation Experiment."
American Journal Of Physics, vol 58, no. 6, 1990, pp. 599-600. American Association
Of Physics Teachers (AAPT), doi:10.1119/1.16412.
Tiersten, Martin S. "Force, Momentum Change, And Motion." American Journal Of Physics,
vol 37, no. 1, 1969, pp. 82-87. American Association Of Physics Teachers (AAPT),
doi:10.1119/1.1975418.
"Conservation Laws - Real-life Applications." Science Clarified.N.p., n.d. Web. Friedl,
Sarah. "Friction: Definition and Types." Study.com.Study.com, n.d. Web.
PhysLink.com, Anton Skorucak. Coefficients of Friction.N.p., n.d. Web.
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