Physics Teaching in Engineering Education PTEE 2009 Institute of Physics, Wroclaw University of Technology, Wroclaw, Poland, September 10-12, 2009

TEACHING COLLISIONS: LABORATORY FRAME VS. CENTER-OF-MASS FRAME

STEFAN NITSOLOV1 and MAYA MITKOVA2 1Department of Applied Physics, Technical University of Sofia

8, Kliment Ohridski St., Sofia-1000, Bulgaria E-mail:sln@tu-sofia.bg

2Department of Mathematics & Natural Sciences, Gulf University for Science and Technology P.O. Box 7207, Hawally 32093, Kuwait

E-mail:mitkovam@gust.edu.kw

A simple and sufficiently comprehensive treatment of central collisions in the introductory physics course for engineers is suggested.

Keywords: teaching collisions, center-of-mass-frame

INTRODUCTION Various aspects of teaching collisions in introductory physics have been discussed

recently [1]-[6]. For engineering students it is advisable to include the coefficient of restitution in the program and to implement the center-of-mass reference frame (C-frame) in the treatment of collisions. Texts in physics for scientists and engineers usually do not consider the coefficient of restitution although it is included into the physics section of requirements for bachelor degree in engineering [7]. A significant improvement of conceptual learning can be achieved by viewing collisions from the C-frame [1], [2], [8], [9]. The advantage is that the use of C-frame reveals the symmetry of momenta, simplifies derivations and clarifies the interpretation of the results. The authors of introductory physics texts use most often only the laboratory frame of reference (L-frame), sometimes frames in which one of the colliding objects is at rest [11] and very rarely the C-frame [10]. A possible reason is that in some texts the definition of center of mass follows the presentation of collisions [12]. Another, and may be the more important reason is that the students have difficulties to form mental image of the impact in the C-frame. To overcome this difficulty we suggest a simple methodology of teaching collisions with the use of C-frame and propose some student activities.

GENERAL PROPERTIES OF COLLISIONS IN C-FRAME

Consider an impact of two bodies with masses 1m and 2m with initial velocities 1v and 2v and final velocities 1v'

and 2v ' , as shown in the figure. We assume the system of colliding

objects is closed and the total momentum is conserved. To relate the initial and final velocities of colliding objects we use a C-frame moving with respect to the L-frame with constant velocity

1 2

1 2

p p PVm m M

(1)

where 1 1 1p m v , 2 2 2p m v

, 1 2P p p and 1 2M m m . The

Physics Teaching in Engineering Education PTEE 2009

velocities u and v of an object measured in the C-frame and L-frame, are related by the Galilean transformation. u v V

(2) The most important advantage of the C-frame is that the total momentum in this frame is

always zero so that the individual momenta are equal and opposite, as shown in the figure. 2 1p p

2 1p p (3)

The velocities of bodies are antiparallel and their magnitudes are inversely proportional to the correspondent masses, as shown below.

1 1 2

2 2 1

u u mu u m

(4)

Conservation of momentum. Velocities in L- and C-frame.

The kinetic energy of the system 2 21 1 2 22 2kE m v m v can be separated into two parts

characterizing the internal motion of the system and the motion of the system as a whole.

2 21 2

1 21 2

12 2k

m mE v v MVm m

(5)

The loss of kinetic energy in a head-on impact is described by the coefficient of restitution ε defined as the ratio of speed of approach to the speed of separation.

1 2

1 2

v vv v

(6)

For elastic collision 1 , for inelastic collisions of first kind 1 , for inelastic collisions of second kind 1 , and for completely inelastic collision 0 .

It is useful following [1] to give the students conceptual examples in which the same head-on impact of two identical balls is viewed from different frames of reference and thus to demonstrate the usefulness of C-frame in the analysis of collisions. In the first example the balls moving with the same speed v undergo a head-on collision. In the second example the first ball moving with speed 2v strikes the second ball initially at rest.

We suggest a third example, shown in the figure. Two identical spheres moving with velocities 1 (2 , 2 )v v v

and 2 (0, 2 )v v

undergo a central elastic collision so that the line of impact is parallel to their relative velocity. It follows from (1) that the velocity of the center of mass is , 2V v v

. Applying transformation (2)

one can obtain that the initial velocities in the C-frame are equal and opposite. Hence, it reduces to the first two examples of the previous paragraph.

Physics Teaching in Engineering Education PTEE 2009

PARTIALLY ELASTIC CENTRAL COLLISIONS In the general case of partially elastic head-on collision, it is convenient to choose the x-axis in the direction of motion and drop the x-subscripts. Combining (4) and (6) we get

1 2 1 2

1 2 1 2

u u u uu u u u

or

1 1u u 2 2u u (7) It is easy to see that the relations between velocities in the C-frame are simpler and easier to interpret than in the L-frame. The last step is to determine the final velocities in L-frame. Using transformation (2) and doing some algebra we have

1 2 1 2 21

1 2

( 1)m m v m vv

m m

2 1 2 1 12

1 2

( 1)m m v m vv

m m

(8)

It should be noted that this derivation is easier than the usual treatment (only in L-frame) of one-dimensional elastic collision in the introductory physics texts [12].

The above results can easily be generalized for central inelastic collisions in two dimensions. For that purpose one has to resolve the velocities in the C-frame into two components: normal (along the line of impact) and tangential (transversal to the line of impact). The equations for each component are identical to (7) with different coefficients of restitution for normal and tangential components. The general tactics however remain the same: solving momentum and coefficient of restitution equations in the C-frame and then using Galilean transformations to transform the results back to the L-frame.

SUGGESTED ACTIVITIES Problem 1. A cylinder of mass 1m shown in the figure slides without rotation on a flat surface with initial speed 1v in the direction of its axis. The side A of the cylinder strikes a resting piston of mass 2m . After the impact the piston starts moving, collides with side B,

recoils back, again collides with A, etc. Assuming that all impacts have the same coefficient of restitution ε and neglecting the friction, find the velocities of the cylinder and piston after n impacts of the piston with the cylinder.

SOLUTION: We choose x-axis in the direction of 1v and drop the subscripts of x-components to simplify notations. The velocities in the C-frame after the first impact are

1 1u u 2 2u u

We see that the velocities of each body before and after the impact are related by the factor , so that the velocities of the bodies after n impacts, ( )

1nu and ( )

2nu , are given by

( )1 1( 1)n n nu u ( )

2 2( 1)n n nu u Finally, using the transformation (2), where 1 21(1 )V v r and 21 2 1r m m , we obtain

21( )1 1

21

1 11

n nn r

v vr

( )2 1

21

1 11

n nnv v

r

Problem 2. Two balls of masses 1m and 2m , 1 2m m , fall together (one on top of the other with small separation) and strike with speed v a fixed horizontal plane, as shown in the figure. If the coefficient of restitution for all impacts is ε, find the ratio of the initial and final speeds of the small ball.

SOLUTION: The collision in the problem is composed of two head-on impacts: first - the massive ball strikes the plane and recoils; second - moving up the

Physics Teaching in Engineering Education PTEE 2009

massive ball collides with the falling small ball. For the first collision the C-frame and L-frame coincide, so that the velocity of the massive ball after the impact with the plane is

2 2 2v u u v The initial velocities of the balls for the second impact are 1v v , 2v v . The centers

of mass of the system and massive ball coincide so that V v . The initial and final velocity of the small ball in the second C-frame are related by the equation (7), 1 1u u .

Using the transformation (2), 1 1u v V and 1 1u v V , we find 1 ( 2)v v or

1

1

( 2)v

Rv

We suggest to show the students a simple and impressive demonstration consisting of the rebound of basketball and baseball dropped above a hard floor. This solution of Problem 2 can be used to explain the results of the demonstration.

Simulation computer programs demonstrating the role of coefficient of restitution [14] and collisions viewed from various reference frames [15] can also be very useful.

CONCLUSIONS In this article, we have emphasized that the use of C-frame provides a concise and straightforward way to teach collisions in the introductory physics course.

References [1] Arons, A., Teaching Introductory Physics. Wiley, (1997) Ch. 1. [2] Loveland, K. T., Simple Equations for Linear Partially Elastic Collisions, The Physics

Teacher, 38 (2000) 380-381. [3] Millet, L. E., The One-Dimensional Elastic Collision Equation: 2 -

f c iv v v , The Physics

Teacher, 36, (1998) 186. [4] Ferreira da Silva, M. F., Meaning and Usefulness of the Coefficient of Restitution,

European Journal of Physics 28 (2007) 1219-1232 [5] Hui Hu, More on One Dimensional Collisions, The Physics Teacher, 40 (2002) 72. [6] Lyublinskaya, I. E., Central Collision – The General Case, The Physics Teacher, 36

(1998) 18-19 [7] NCEES, Fundamentals of Engineering, Supplied-Reference Handbook, 5/E (2001) p. 26. [8] Gregory, R. D., Classical Mechanics, Cambridge University Press (2006) Ch. 10. [9] Arya, A. P., Introduction to Classical Mechanics, 2/E, Addison-Wesley (1997) Ch. 8. [10] Sivuhin, D. V., General Physics Course, Vol. 1, Mechanics, Moscow (1979), (in

Russian), pp. 151-152. [11] Knight, R. D., Physics for Scientists and Engineers: A Strategic Approach, 2/E,

Pearson Education International (2008) Ch. 10. [12] Fishbane, P. M., S. G. Gasiorowicz, S. T. Thornton, Physics for Scientists and

Engineers, 3/E, Pearson Education International (2005) Ch. 8. [13] Serway, R. A., J. W. Jewett, Physics for Scientists and Engineers, Thomson

Brooks/Cole (2004) Ch. 9. [14] Fu-Kwun Hwang, Bouncing Ball, Webpage of Department of Physics, National

Taiwan Normal University, Web address: http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=345

[15] Fu-Kwun Hwang, 1D collision: Conservation of momentum, Webpage of Department of Physics, National Taiwan Normal University, Web address: http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=5