Environmental Engineering Reference
In-Depth Information
6.5
γ
=
3
.
20 and hence length is 31 cm.
6.6
γ
4
.
11 and hence distance is 2.9 km.
6.7 In a spaceship, the 10
5
light years becomes length contracted. Equivalently,
a journey time of order 10
5
Earth years can be reduced by time dilation.
Suppose we want the journey to take just 20 years. Then we require that the
speed relative to Earth,
u
, should satisfy
=
10
5
c
1
20
u
=
−
u
2
/c
2
.
Solving gives
u
0
.
99999998
c
. Note you can avoid solving a quadratic by
realising that
γ
is very large and hence
u
is very close to
c
so it is a good
approximation to solve
γ
=
10
5
/
20 for
u
giving
u/c
10
−
8
.
=
≈
1
−
2
×
10
7
)
6.8 (a)
γ
=
1
.
009 hence length is 3.57 km. (b) 3600
/(
4
.
00
×
=
9
.
00
×
10
−
5
10
7
)
10
−
5
s. (c) 3570
/(
4
.
00
×
=
8
.
92
×
s.
6.9 (a) Length measured by friend is 10
/γ
6 m. (b) Since pole is 2 m short
of the barn length the time delay is 2 m
/(
0
.
8
c)
=
10
−
9
s. (c) Now
the barn is contracted to a length of 4.8 m whilst the pole remains at 10 m.
(d) From the athlete's viewpoint, the rear of the pole cannot know that the
front has struck the wall until at least 10 m
/c
=
8
.
33
×
10
−
8
s after it has
done so (since no signal can travel faster than the speed of light). In this
time interval, the barn door can travel “for free” a distance of 10 m
/c
=
3
.
33
×
×
0
.
8
8 m which is plenty long enough for the pole to fit within the barn. Thus
the apparent paradox is resolved.
6.10 Use
c
=
1
1
/
2
−
v/c
λ
=
λ
0
1
+
v/c
with
λ
0
=
589 nm and
λ
=
550 nm to give
v/c
=
0
.
0684.
1
.
5km.Wearealsotoldthat
t
A
−
t
B
=
6.11
t
B
−
0,
where the primes indicate times measured in the observer's rest frame. Need
to identify the relevant Lorentz transformation formula and the most useful
one is the one involving the given quantities, i.e. we use “
t
=
t
A
=
1
.
3
µ
sand
x
B
−
x
A
=
vx/c
2
)
”.
Subtracting the equation pertaining to event A from that pertaining to event
Bgives
γ(t
−
γ
1
.
3
µ
s
1500 m
v
c
2
0
=
−
and hence
v
=
0
.
26
c
.
4sand
t
B
−
t
A
=
6.12 Given
x
B
−
x
A
=
0,
t
B
−
t
A
=
5 s we are asked to deduce
x
B
−
x
A
. The relevant Lorentz transformation formula states that
t
=
γ
t
vx/c
2
where
x
−
=
x
B
−
x
A
etc. Putting the numbers in gives
=
=
γ
0
.
6. Note that you could get this directly from the
time dilation formula since the events occur at the same place in one of the
frames. The spatial separation is 5 s
5
/
4 hence
v/c
10
8
×
0
.
6
c
=
9
.
0
×
m.
1kmand
x
B
−
x
A
=
6.13
t
B
−
t
A
=
0,
x
B
−
x
A
=
2kmweareaskedtodeduce
t
B
−
t
A
. The relevant Lorentz transformation formula states that
x
=
γ
