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Teaching Collisions - Methodological Suggestions
STEFAN NITSOLOV, MAYA MITKOVA*
Department of Applied Physics
Technical University of Sofia
boul. Kliment Ohridski 8, Sofia 1000
BULGARIA
*Department of Mathematics and Natural Sciences
Gulf University for Science and Technology
P.O. Box 7207 Hawally 32093
KUWAIT
Abstract: - This paper suggests some changes in the content and methodology of teaching collisions. It is
stimulated by the research in the related area and authors’ teaching experience. The traditional approach of
teaching collisions in the introductory university physics course does not reveal enough the nature of
phenomenon and thus limits the functional understanding of students. Authors propose to implement center of
mass frame of reference and coefficient of restitution in the treatment of collisions. The presented
methodological approach aims to clarify the physics of the phenomenon and to make it more comprehensible.
Key-Words: collision, methodology, center of mass, introductory physics, engineering education
1 Introduction Suggestions for improvement of teaching
collisions were expressed by number of authors.
Arons [1], following Huigens, recommends to view
the impact from different frames of reference. The
advantage of vector representation of the velocities
before and after the impact in the center of mass
reference frame (c.m. frame) is pointed out by
Lyublinskaya [1]. Millet [3] proposes to replace
traditional textbook equations for the final velocities
after the impact with simpler equations including
the velocity of the center of mass. Loveland [4]
recommends to derive formulas for the final
velocities in inelastic head-on collision using c.m.
frame and coefficient of restitution. Similar
considerations are made by Hui Hu [5]. The
importance of the coefficient of restitution for
description of losses of energy in collisions has been
discussed in the work of da Silva [6]. We have to
mention also: teaching materials [7], [8] in the
Internet in which the collisions are treated using
c.m. frame and coefficient of restitution; computer
programs, which demonstrate the role of coefficient
of restitution [8] and visualize collisions viewed
from various reference frames [10]; laboratory
exercise in which the students analyze 2D collisions
both in lab frame and c.m. frame [11].
Unfortunately, the authors of introductory
physics texts use most often only the lab frame,
sometimes frames in which one of the colliding
objects is initially at rest [12] and very rarely c.m.
frame. A possible reason is that in some texts [13],
[14] the definition of center of mass follows the
presentation of collisions. Another, and may be
more important reason is that the students do not
have enough experience to transfer observations
between two different reference frames and the fact
that the c.m. frame is associated with an abstract
point, not a real reference body. At the same time
treating collisions in the c.m. frame instead of lab
frame has some important advantages - discloses the
symmetry of momenta, simplifies derivations, and
clarifies the interpretation of the results.
Most textbooks consider only perfectly elastic or
inelastic impacts reducing by this the whole variety
of real collisions to two idealized cases. In this
connection we recommend to incorporate the
coefficient of restitution in the topic content. It is
quite necessary especially for engineering students
(the coefficient of restitution is included in the
physics section of requirements for the USA
bachelor degree in engineering [15]).
As we mentioned above the traditional textbook
approach in teaching collisions limits the conceptual
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understanding of students. This limitation is well
seen in solving problems – students usually are able
to solve problems only by memorizing the formulas
for the final velocities of colliding bodies and using
“plug and chug” method. Looking for better
approach, we suggest a simple methodology of
teaching collisions with the use of c.m. frame and
coefficient of restitution.
2 Central Collisions in the Center of
Mass Frame: Definitions and General
Properties Collisions are central when the centers of mass
of colliding bodies lie on the line of impact, which
is the common normal to the contacting surfaces
passing through the contact point. For example
collisions of balls are always central. Consider the
case of central collision in three dimensions of two
objects of masses 1m and
2m , viewed from the lab
frame (Fig. 1).
Fig. 1 Two body collision viewed from the lab
frame.
Let 1v
and 2v
be initial velocities and 1v'
and2v '
be the velocities of the bodies after collision. We
assume the system of colliding objects is closed and
the total momentum is conserved (the students
should already be familiar with the laws of
conservation of momentum and energy). If the
center of mass is not introduced yet, it has to be
defined here. The position vector of the center of
mass C of the system of two bodies (with position
vectors 1r
and 2r
) is given by the equation
1 1 2 2
1 2
m r m rR
m m
(1)
The center of mass moves with respect to the lab
frame with constant velocity
1 1 2 2
1 2
m v m vdR PV
dt m m M
(2)
where1 11 2 2P m v m v
and
1 2M m m are the
total momentum and mass of the system,
respectively. This follows from the law of
conservation of momentum. The velocities u
and
v
of an object measured in the c.m. frame and lab
frame are related by the Galilean transformation
u v V
(3)
We use the c.m. frame because it possess an
important property – the total momentum in this
frame is always zero (it can be shown setting 0V
in Eq. (2)), so that the individual momenta are equal
and opposite before and after collision.
2 2 1 1m u m u
2 2 1 1m u m u
(4)
The momenta in the lab frame and c.m. frame are
shown in Fig. 2. From Eq. (4) it follows that at any
instant the velocities of the two bodies are
antiparallel and their magnitudes are inversely
proportional to their masses. 1 1 2
2 2 1
u u m
u u m
(5)
Fig. 2 Momenta of colliding bodies in: a) the lab
frame; b) the c. m. frame
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Equations (4) for momenta and (5) for speeds are
simple and useful but in general not enough to
express the final velocities when given the initial, or
vice versa. For this purpose it is necessary to add
equations for the energy. The kinetic energy of the
system
2 2
1 1 2 2
2 2k
m v m vE can be written in
equivalent form (using Eq. (1) and some tedious
algebra).
2 21 21 2
1 2
1
2 2k
m mE v v MV
m m
(6)
The first part is the energy of internal motion of
the system which changes when the relative speed
1 2v v
of bodies varies. The second part which is
the energy of motion of the system as a whole
remains constant during collision, and is equal to
zero in the c.m. frame, where 0V
.
The change of relative speed in collisions, which
is equivalent to change in kinetic energy, is
characterized by the coefficient of restitution. In a
direct central (head-on) impact the coefficient of
restitution is defined as the ratio of speed of
approach to the speed of separation of bodies.
2 1
2 1
v v
v v
(7)
The relation between the loss of energy in
collision and coefficient of restitution can be made
clear by calculating the change in kinetic energy in
the c.m. frame from Eq. (6) and (7).
2 1k k kE E E (8)
Equation (8) helps to understand better the
classification of collisions in terms of energy: 1
- elastic collision; 1 - inelastic collisions of first
kind (loss of kinetic energy); 1 - inelastic
collisions of second kind (gain of kinetic energy);
0 - completely inelastic collision.
3 Derivation of the Final Velocities in
Central Collision To demonstrate the usefulness of c.m. frame in
the analysis of collisions we recommend to discuss
with the students the following conceptual example.
Two identical balls of mass m undergo head-on
inelastic collision with coefficient of restitution .
In the lab frame, the first ball is approaching with
given velocity v
and the second ball is at rest. (Fig.
3 a)).
Fig. 3 Collision of identical balls viewed from:
a) the lab frame and b) the c. m. frame.
The instructor may ask the students to determine the
initial and final velocities in the c.m. frame and the
final velocity in the lab frame. Since all velocities
are directed along the line of impact, vector
notations can be dropped. From Eq. (2) and (3) one
can find 2
vV and
1 22
vu u . The collision
viewed from the c.m. frame is shown in Fig. 3 b).
Next it must be pointed out that in the c.m. frame
the collision is symmetrical which should lead the
students to the conclusion that the final velocities
should also be equal and opposite, 1 2u u . Using
Eq. (7) one can find the final velocities in the c.m.
frame and transform them to the lab frame by using
Eq.(3).
1
1
2v v
2
1
2v v
The instructor assigns the same problem to be
solved in the lab frame pointing out that the
advantage of the use of c.m. frame is more obvious
in the case of more complicated problems.
To derive the final velocities in the general case
of direct central collision, one can rewrite (5)
dropping vector notations and using the properties
of proportions in the form 1 2 1 2
1 2 1 2
u u u u
u u u u
.
Combining this with Eq. (7) we obtain
1 1u u 2 2u u . (9)
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The last step is to transform Eq.(9) from the c.m.
frame to the lab frame using Eq. (2) and (3).
1 2 1 2 2
1 1
1 2
2 1 2 1 1
2 2
1 2
( 1)( 1)
( 1)( 1)
m m v m vv v V
m mm m v m v
v v Vm m
(10)
Comparing Eq. (9) and (10) one can see that in
the c.m. frame the relations between the velocities
of colliding bodies are simpler and the interpretation
is easier than in the lab frame.
The above results can be generalized for oblique
(in three dimensions) central collisions. For that
purpose one has to resolve the velocities in the c.m.
frame into two components: normal (along the line
of impact) and tangential (perpendicular to the line
of impact). The equations for the normal
components are identical to Eq.(9). In the special
case of smooth bodies the tangential components do
not change during the collision. When the bodies are
not smooth, the tangential contact forces give rise to
a change in tangential components of relative
velocity described by restitution relation similar to
Eq.(7). The general tactics remains the same:
solving momentum and coefficient of restitution
equations in the c.m. frame and then using Eq. (3) to
transform the results back to the lab frame.
4 Summary In this article, we have emphasized the use of
c.m. frame and coefficient of restitution. This
provides a concise and straightforward way to teach
collisions in the introductory physics course. The
general equations for central collision are simpler in
the c.m. frame than in the lab frame. Employing
multiple presentations of the same impact in
different reference frames helps the students to
move from novice approach to a more expert-like
problem-solving approach. The difficulties
associated with the transition between different
frames of reference are compensated by the
improvement of students’ conceptual understanding.
References:
[1] A. Arons, Teaching Introductory Physics, (New York, John Wiley & Sons, 1997) pp.
8-14.
[2] I. E. Lyublinskaya, Central Collision – The
General Case, Phys. Teach., 36, 18-19 (January
1998).
[3] L. E. Millet, The One-Dimensional Elastic
Collision Equation: 2 -f c iv v v , Phys. Teach.,
36, 186 (March 1998).
[4] K. T. Loveland, Simple Equations for Linear
Partially Elastic Collisions, Phys. Teach., 38,
380-381 (Sept. 2000).
[5] Hui Hu, More on One Dimensional Collisions,
Phys. Teach., 40, 72 (Feb. 2002).
[6] Ferreira da Silva, M. F., Meaning and
Usefulness of the Coefficient of Restitution,
Eur. J. Phys., 28, pp. 1219-1232 (Oct. 2007).
[7] P. Hunter, Elastic Collisions: Zero Total
Momentum:
http://www.tacomacc.edu/HOME/phunter/Elast
ic%20Collisions%20COM.pdf
[8] R. Vawter, Department of Physics and
Astronomy, Western Washington University,
http://www.ac.wwu.edu/~vawter/PhysicsNet/T
opics/TopicsMainTemplate.html
[9] Simulation of the collision of two particles,
University of Florida Distance Continuing &
Executive Education:
http://vam.anest.ufl.edu/physics/collisionphysic
s.html
[10] 1D collision: Conservation of momentum,
Department of Physics, National Taiwan
Normal University:
http://www.phy.ntnu.edu.tw/ntnujava/index.ph
p?topic=5
[11] D. Brown A. Cox, Innovative uses of video
analysis, Phys. Teach., 47, pp. 145-150 (March
2009).
[12] R. D. Knight, Physics for Scientists and
Engineers: A Strategic Approach, 2nd ed.
(Pearson Education International, 2008) Ch. 10.
[13] P. M. Fishbane, S. G. Gasiorowicz, and S. T.
Thornton, Physics for Scientists and Engineers,
3rd ed. (Pearson Addison-Wesley, San
Francisco, 2005) pp. 210-245.
[14] J. Jewett and R. Serway, Physics for Scientists
and Engineers with Modern Physics, 7th ed.
(Thomson Learning, Belmont, 2008), Ch. 9.
[15] NCEES, Fundamentals of Engineering,
Supplied-Reference Handbook, 5th ed.
(NCEES, Clemson, 2001) p. 26.X1.
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