bei48482_FM

22 Chapter One

1.7 RELATIVISTIC MOMENTUM

Redefining an important quantity

In classical mechanics linear momentum pmvis a useful quantity because it is con-

served in a system of particles not acted upon by outside forces. When an event such

as a collision or an explosion occurs inside an isolated system, the vector sum of the

momenta of its particles before the event is equal to their vector sum afterward. We

now have to ask whether pmvis valid as the definition of momentum in inertial

frames in relative motion, and if not, what a relativistically correct definition is.

To start with, we require that pbe conserved in a collision for all observers in rel-

ative motion at constant velocity. Also, we know that pmvholds in classical

mechanics, that is, for c. Whatever the relativistically correct pis, then, it must

reduce to mvfor such velocities.

Let us consider an elastic collision (that is, a collision in which kinetic energy is

conserved) between two particles Aand B, as witnessed by observers in the reference

frames Sand S which are in uniform relative motion. The properties of Aand Bare

identical when determined in reference frames in which they are at rest. The frames S

and S are oriented as in Fig. 1.13, with S moving in thexdirection with respect

to Sat the velocity v.

Before the collision, particle Ahad been at rest in frame Sand particle Bin frame

S. Then, at the same instant, Awas thrown in the ydirection at the speed VAwhile

Bwas thrown in the y direction at the speed VB , where

VAV (^) B (1.10)
Hence the behavior of Aas seen from Sis exactly the same as the behavior of Bas seen
from S.
When the two particles collide, Arebounds in the ydirection at the speed VA,
while Brebounds in the y direction at the speed VB. If the particles are thrown from
positions Yapart, an observer in Sfinds that the collision occurs at y^12 Y and one in
S finds that it occurs at y y^12 Y. The round-trip time T 0 for Aas measured in
frame Sis therefore
T 0  (1.11)
and it is the same for Bin S :
T 0 
In Sthe speed VBis found from
VB (1.12)
where Tis the time required for Bto make its round trip as measured in S. In S , however,
B’s trip requires the time T 0 , where
T (1.13)
T 0

 1 ^2 c^2
Y

T
Y

V (^) B
Y

VA
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