bei48482_FM

24 Chapter One

From Eq. (1.11), VA

If we use the classical definition of momentum, pmv, then in frame S

pAmAVAmA

pBmBVBmB  1 ^2 c^2 

This means that, in this frame, momentum will not be conserved if mAmB,where

mAand mBare the masses as measured in S. However, if

mB (1.14)

then momentum willbe conserved.

In the collision of Fig. 1.13 both Aand Bare moving in both frames. Suppose now

that VAand V (^) Bare very small compared with , the relative velocity of the two frames.
In this case an observer in Swill see Bapproach Awith the velocity , make a glanc-
ing collision (since VB ), and then continue on. In the limit of VA0, if mis the
mass in Sof Awhen Ais at rest, then mAm. In the limit of VB 0, if m() is the
mass in Sof B,which is moving at the velocity ,then mBm(). Hence Eq. (1.14)
becomes
m() (1.15)
We can see that if linear momentum is defined as
p (1.16)
then conservation of momentum is valid in special relativity. When c, Eq. (1.16)
becomes just pmv, the classical momentum, as required. Equation (1.16) is often
written as
pmv (1.17)
where
 (1.18)
In this definition, mis the proper mass(or rest mass) of an object, its mass when
measured at rest relative to an observer. (The symbol is the Greek letter gamma.)
1

 1 ^2 c^2
Relativistic
momentum
mv

 1 ^2 c^2
Relativistic
momentum
m

 1 ^2 c^2
mA

 1 ^2 c^2
Y

T 0
Y

T 0
Y

T 0
bei48482_ch01.qxd 1/15/02 1:21 AM Page 24