Environmental Engineering Reference
In-Depth Information
scattering process would in principle suffice to determine
f(v/c)
, it is only that
this process provides a particularly elegant path to the answer. Having said that,
the derivation we shall now present is still rather tricky and any readers wishing to
avoid the details might note the key results presented in Eq. (7.15) and Eq. (7.25)
and skip to the next subsection. We shall return to the topic of energy and momen-
tum in Part IV where we shall see that there is a much more elegant way to obtain
the results which we shall here work rather hard to establish.
Returning to Figure 7.1, we consider the special case where particle 1 travels
in to and out of the scattering at the same angle
α
to the
x
axis, and particle 2
travels always parallel to the
y
axis with speed
v
. We shall also assume that the
particles have the same mass
m
. Momentum conservation in the
x
direction then
implies that if the speed of particle 1 is
u
before the scatter then it should remain
unchanged after the scatter. This is a very symmetrical scattering and it is this
symmetry which will help us arrive at a form for
f(v/c)
without too much hard
work. In order to simplify matters still further, we focus our attention upon the
limit that
α
0 too since otherwise the process
would not conserve momentum in the
y
direction. This limit will help us a great
deal since it will allow us to use the non-relativistic form for the momentum of
particle 2, i.e. we shall make use of
f(
0
)
→
0. In this limit the speed
v
→
=
1. We have now completely specified
the scattering process in inertial frame
S
and it clearly conserves momentum in the
x
and
y
directions independent of the actual form of
f(v/c)
.
Now let us view the same scattering from a second inertial frame,
S
. In particular
we choose a frame which travels along the positive
x
axis at a speed
u
cos
α
.In
this frame, particle 1 travels only in the
y
direction, whilst particle 2 travels in
the negative
x
direction as illustrated in Figure 7.2. The velocity addition formulae
allow us to write down the
x
and
y
components of velocity of each particle in
S
:
v
1
x
=
0
,
1
u
sin
α
v
1
y
=−
(u/c)
2
cos
2
α)
−
1
/
2
(u/c)
2
cos
2
α
(
1
−
1
−
u
sin
α
=−
1
(u/c)
2
cos
2
α
,
(7.9)
−
y'
S'
u
cos a
1
u
1
y
= −u
a
a
u
2
y
=
u
sin a
2
u
2
x
=
−
u
cos a
x'
Figure 7.2 Particle 2 scatters through a vanishingly small angle
α
, whilst particle 1 bounces
off at right angles.
