Environmental Engineering Reference
In-Depth Information
distance than L ,i.e.
ct in =
L
vt in
L
t in =
v .
(6.13)
c
+
Adding together Eqs. (6.12) and (6.13) and equating the result to Eq. (6.10) gives
an equation relating L and L 0 ,i.e.
L
L
γ 2 L 0
c
v +
=
.
(6.14)
c
+
c
v
Solving for L gives
L 0
γ
=
L
.
(6.15)
Again a remarkable result; for the length of the ruler is smaller when it is in
motion than when it is at rest.
We could have anticipated the length contraction result knowing only the time
dilation result. The argument goes as follows. Let us consider again the muons
created in the upper atmosphere which we discussed in Example 6.1.1. From the
viewpoint of a muon, it still lives for 2 . 2
s yet has travelled all the way to the
Earth's surface. However this is not such an impossible task as it would be in
classical theory for the 20 km is reduced by a factor of γ . It has to be exactly the
same factor of γ as before because we know that muons created at an altitude of
20 km on average just reach the Earth before decaying if they have a speed of 0 . 999 c
and from the viewpoint of such a muon the Earth moves towards it at that speed.
Example 6.1.3 A spaceship flies past the Earth at a speed of 0 . 990 c. A crew member
on the ship measures its length to be 400m. How long is the ship as measured by
an observer on Earth?
µ
Solution 6.1.3 This is a straightforward application of the length contraction result
expressed in Eq. (6.15) with L 0 =
400 m. Hence
1
γ
=
1
=
7 . 09
(6.16)
0 . 990 2
and so L
56 . 4 m. Perhaps the most common misuse of the length
contraction formula is to confuse L and L 0 .
=
400 / 7 . 09
=
6.1.3 Simultaneity
Classical physics, with its absolute time, has an unambiguous notion of what it
means to say two events are simultaneous. However, since time is more subjective
in Special Relativity, having meaning only within the context of a specified inertial
frame, it may not be suprising to hear that two events that are simultaneous in
one inertial frame will not in general be simultaneous in another inertial frame.
Moreover, according to one observer event A may precede event B but according to
a second observer event B might occur first. This last statement sounds particularly
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