Environmental Engineering Reference
In-Depth Information
Solution 6.1.1
(a) Muons travelling at speed c will (on average) travel, according
to classical thinking, a distance ct
0
before decaying where t
0
=
2
.
2
µ
s. Putting
the numbers in gives a distance of just 660m.
(b) Let us suppose that the muon is travelling at a speed u towards the Earth. In
the muon's rest frame its lifetime is a mere t
0
=
2
.
2
µ
s but from the point of view
of an observer on Earth this lifetime is dilated to t
γt
0
.Ifγ is sufficiently
large it is therefore possible that the muon could travel the 20 km and reach the
Earth's surface. We can determine how large u must be using
=
20
km
u
γt
0
>
.
(6.7)
u
2
/c
2
)
−
1
/
2
we can solve this equation for u
Since γ
0
.
999
c. Today, the
lifetime of the muon has been measured as a function of its speed and it is found to
be in excellent agreement with the prediction of time dilation.
=
(
1
−
=
Before leaving our discussion of time dilation we pause to consider the situation
illustrated in Figure 6.3. Figure 6.3(a) shows our two frames
S
and
S
moving
relative to each other as shown. Time dilation says that, according to an observer
at rest in
S
,clocksin
S
run slow, i.e. that
t
γt
. This really does mean
that all clocks run slow and so according to
S
an observer in
S
would age more
slowly. Now consider Figure 6.3(b). It represents exactly the same situation as
Figure 6.3(a) since one can either think of
S
=
moving relative to
S
or vice versa.
Now an observer in
S
will conclude that clocks in
S
run slow, i.e. that
t
=
γt
and so from their perspective an observer at rest in
S
would age more slowly.
At first glance these two conclusions seem to contradict each other but they do
not since the observers are measuring intervals of time between different pairs
of events: the observer in
S
is using clocks at rest in
S
whereas the observer in
S
′
S
z
z
′
u
y
′
y
x
′
O
x
O
′
(a)
z
S
z
′
S
′
u
y
y
′
x
x
′
O
O
′
(b)
Figure 6.3 Two observers each conclude that the other is ageing more slowly than them-
selves. This is not a contradiction.
