29-9:
Picture the Problem
: Due to time dilation, as a proper time of 1.00 seconds pass on the ship, 1.00 minute
passes on Earth.
Strategy:
The two events to be considered in this case are successive ticks of the clock that is in your
spaceship.
From your point of view, these events occur at the same place, so you observe the proper time
interval
t
0
and an observer on Earth measures the dilated time interval
t
. Solve equation 29-2 for the
speed of the ship, where
and
0
1.0000 s
t
1.0000 m = 60.000 s.
t
Solution
: 1. Solve equation 29-2 for the speed:
0
0
2
2
1
1
t
t
v
c
v
c
t
t
2.
Insert the two time intervals:
2
1.0000 s
1
0.99986
60.000 s
v
c
c
Insight:
If 1.0000 hour had passed according to an Earth observer, instead of 1.0000 minute, the speed
would have to be much closer to the speed of light. The speed would have been 0.9999999614
c
.
29-26:
Picture the Problem
: The image shows a rectangular
painting as viewed in its rest frame. The painting travels at
sufficient speed that a stationary observer views the
painting as square.
Strategy:
The height of the painting will not contract
because that dimension is perpendicular to the motion.
Therefore, use equation 29-3 to calculate the painting’s
speed at which its length will be measured as 80.5 cm.
Solution:
1.
Solve equation 29-3 for the speed
v
:
2
2
2
2
0
0
1
1
L
L
v
c
v
c
L
L
2.
Substitute numerical values:
2
1
80.5 cm 124 cm
0.761
v
c
c
Insight:
It is not possible to make the painting square by moving the painting in the vertical direction,
because length contraction would only shorten that dimension.
29-41:
Picture the Problem
: The image shows two ships
(1 and 2) heading toward Earth (3) at speeds
and
relative to Earth.
13
0.80
v
c
23
0.80
v
c
Strategy:
We want to calculate the speed of the
one ship relative to the other. Use equation 29-4 to
calculate
the speed of Ship A relative to Ship
B.
12
,
v
Solution:
1.
Multiply both sides of
equation 29-4 by the denominator:
12
23
13
2
12
23
2
13
12
23
12
23
1
1
v
v
v
v v
c
v
v v
c
v
v
2.
Isolate the unknown
variable
and solve:
12
v
2
12
23
13
13
23
13
23
12
2
2
23
13
1
0.80
0.80
1
1
0.80
0.80
v
v v
c
v
v
c
c
v
v
v
v v
c
c
c
c

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