Environmental Engineering Reference
In-Depth Information
y
S
A
C
B
D
x
→
Figure 7.3
A general two-to-two scattering process: AB
CD.
u
B
. These particles subsequently scatter into particles C (mass
m
C
)andD(mass
m
D
) travelling with velocities
u
D
. We say that the scattering process is
a two-to-two process and we denote it AB
u
C
and
CD. It should be clear that the
discussion which now follows can be generalised to include more incoming and/or
outgoing particles without too much trouble.
We want to check that if momentum is conserved in
S
,i.e.
→
p
A
+
p
B
=
p
C
+
p
D
(7.16)
then
p
A
+
p
B
=
p
C
+
p
D
(7.17)
also holds, where the primes indicate the momenta are appropriate to the
S
frame
of reference, which moves along the positive
x
axis with a speed
U
. Apart from
providing us with the confidence that Eq. (7.15) is not an accident of choosing
the symmetric scattering process discussed above, this check has the bonus that it
will also indicate the way in which we should modify the law of conservation of
energy. In order to proceed, we are going to resolve the momenta into their
x
and
y
components. Let us start with the
y
components first (they are a little easier to
deal with). Our strategy is to check directly Eq. (7.17). To do this we need to know
how each of the momenta in
S
can be expressed in terms of their components in
S
. For particle A we have
2
,
p
Ay
γ(v
A
)m
A
v
Ay
=
1
γ(U)
v
Ay
γ(v
A
)m
A
·
=
(7.18)
1
−
Uv
Ax
/c
2
and to get the second line we used the velocity addition formula to relate the
y
component of the velocity in
S
to that in
S
. We need to relate
γ(v
A
)
to quantities
defined in
S
. This is where the hard work resides, if you are prepared to trust our
2
The same methods can be used for each of the other particles.
