Environmental Engineering Reference
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and using the fact that momentum is conserved (i.e. m 1 u 1 +
m 2 u 2 =
m 1 v 1 +
m 2 v 2 )
this can be written as
1
2 m 1 u 1
1
2 m 2 u 2
1
2 m 1 v 1
1
2 m 2 v 2 .
+
=
+
(7.6)
Crucial to this argument is the fact that the law of addition of velocities is linear.
However, we know that the relationship in Special Relativity is non-linear. For
example,
u 1
U
u 1 =
(7.7)
u 1 U/c 2
1
and the factor of u 1 in the denominator spoils the linearity. We are therefore forced
to seek out alternatives to Eq. (7.1) and Eq. (7.2) which do satisfy the 1 st postulate.
Let us aim first for a new definition of momentum. Insisting that it remain a
vector quantity that lies parallel to the velocity and which is equal to the classical
result in the limit v
c dictates that the most general form available to us is
p =
v
f(v/c)m
(7.8)
and our task is to determine the dimensionless function 1
f(v/c) which we know
must satisfy f(v/c)
c . The mass of the particle is labelled m and we
take it to be an intrinsic property of the particle, not depending upon its state of
motion, i.e. all observers will agree upon its value. To determine f(v/c) we are
going to focus our attention upon the very specific scattering process illustrated
in Figure 7.1. Our strategy is to view this process in two different inertial frames
with the momentum carefully defined so that it is conserved in both frames. Any
1for v
y
S
1
u
a
a
u
2
x
Figure 7.1 Particle 1 scatters through a vanishingly small angle α , whilst particle 2 bounces
off at right angles.
1 We write the argument explicitly as v/c to remind us that this is the only way to form a dimensionless
function of the speed.
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