Environmental Engineering Reference
In-Depth Information
and
v
2
x
=−
u
cos
α,
1
v
2
y
=
(u/c)
2
cos
2
α)
−
1
/
2
v
(
1
−
v
1
(u/c)
2
cos
2
α.
=
−
(7.10)
It is now that we can appreciate the advantage of picking such a symmetric scat-
tering process. Viewed in
S
, the scattering looks just like that in
S
except that it
is 'turned upside down'. This symmetry allows us to conclude that
v
1
y
=−
v
and
v
2
y
=
u
sin
α.
(7.11)
Either of these two equations used in conjunction with Eq. (7.9) or Eq. (7.10)
implies that
u
sin
α
v
=
1
.
(7.12)
(u/c)
2
cos
2
α
−
Now we require momentum conservation in
S
,i.e.
=
2
mf (u/c)v
2
y
2
mf (v/c)v.
But in the limit of
α
→
0 we can safely take
f(v/c)
→
1, i.e.
f(u/c)u
sin
α
=
v.
(7.13)
Subsituting in for
v
using Eq. (7.12) gives
u
sin
α
f(u/c)u
sin
α
=
1
,
(u/c)
2
cos
2
α
−
which in the limit
α
→
0 reduces to
1
f(u/c)
=
1
=
γ(u).
(7.14)
−
u
2
/c
2
Thus we have a new candidate for momentum in Einstein's theory. For a particle
of mass
m
and velocity
u
the momentum is
p
=
γ(u)m
u
.
(7.15)
Although we have a candidate for momentum, we ought to be clear that we
derived it by considering one very particular scattering process. If this definition is
to be useful then it ought to have the property that if momentum is conserved in one
inertial frame then it is also conserved in all other inertial frames and this should
be true for any process. To convince ourselves that this is the case, let's consider
the much more general scattering process illustrated in Figure 7.3. In inertial frame
S
, particles A (mass
m
A
)andB(mass
m
B
) are incident with velocities
u
A
and
