Environmental Engineering Reference
In-Depth Information
1
/
1
Using γ
=
−
v
2
/c
2
gives us an expression for the speed:
v
2
c
2
1
γ
2
.
=
1
−
(a) For the TV set,
10
3
eV
mc
2
20
.
0
×
1
+
=
1
+
=
1
.
039
.
511
×
10
3
Hence the speed is given by
1
v
c
1
1
.
039
2
=
−
=
0
.
272
.
It is usually most convenient to express speeds as a fraction of the speed of
light.
(b) For the X-ray machine,
10
6
eV
mc
2
5
.
00
×
1
+
=
1
+
=
10
.
8
511
×
10
3
and the speed is therefore
1
v
c
=
1
10
.
8
2
−
=
0
.
996
.
7.2.1 When is relativity important?
c
and the formulae of
non-relativistic mechanics provide a good approximation. Usually it makes sense
to use the non-relativistic approach if one can be confident that it provides
sufficient accuracy since it is usually easier than computing using the full
apparatus of Special Relativity. For example, one really can safely neglect
relativistic corrections when building a car (except for the satellite navigation
system which uses the Global Positioning System (GPS)). It would certainly be an
advantage if we could spot whether or not a system needs relativistic corrections
before performing the necessary calculations. Clearly if we know that a particle
is travelling with speed much smaller than the speed of light then we can press
ahead using Newton's mechanics. But what if we are given the kinetic energy or
the total energy of a particle, is there a quick way to tell if it is relativistic or not?
The answer is of course in the affirmative: if a particle has a kinetic energy
which is much smaller than its rest mass energy then the particle is moving
non-relativistically whereas if the kinetic energy is comparable to or greater than
the rest mass energy the particle is moving relativistically. To see this we need to
realise that the non-relativistic limit corresponds to
γ
We
know
that
when
γ
1
it
follows
that
v
1 in which case the kinetic
1
)mc
2
is much smaller than the total energy
γ mc
2
. Another way of
energy
(γ
−
