2.1 Find the value of g for u 5 (0.84)1/2c, u 5 0.6c, u 5 0.8c, and u = 0.866c.
Solution:
From Equations 2.5 and 2.7 we have
so direct substitution for u gives
2.2 Express g as a series, using the binomial expansion.
Answer: g 5 1 1 b21 b41 ...
2.3 Two inertial systems are receding from one another at a uniform speed of 0.6c. In one system a sprinter runs 200 m in 20 s, according to his stop-watch. If the path of the sprinter is perpendicular to the axis of relative motion between the two systems, how far did the sprinter run and how
5 9
long did it take him, according to observers in the other inertial system?
Solution:
Given u = 0.6c, Dz9 5 200 m, Dt9 5 20 s, and Dz 5 Dz9 5 200 m, ac-cording to Equation 2.2. With b 5 u/c 5 0.6, g can be computed with Equation 2.7,
and Equation 2.8 immediately yields
Dt 5 gDt9 5 ( ) (20 s) 5 25 s.
2.4 Consider Problem 2.3 for the situation where the path of the sprinter is parallel to the axis of relative motion between the two systems.
Answer: 160 m, 25 s
2.5 An observer moves at 0.8c parallel to the edge of a cube having a proper volume of 153cm3. What does the observer measure for the volume of the cube?
Solution:
With u = 0.8c → g 5 5/3, Dy 5 Dy9 5 15 cm, Dz 5 Dz9 5 15 cm, and Dx9 5 15 cm,
Accordingly, the volume measured by the observer is V 5 DxDyDz 5 (9 cm)(15 cm)(15 cm) 5 2025 cm3.
2.6 Two inertial systems are uniformly separating at a speed of exactly Ï0.84wc. In one system a jogger runs a mile (1609m) in 6 min along the axis of relative motion. How far in meters does he run and how long does it take, to observers in the other system?
Answer: 643.6 m, 15 min
2.7 Consider two inertial systems separating at the uniform speed of 3c/5.
If a rod is parallel to the axis of relative motion and measures 1.5 m in its system, what is its length to observers in the other system?
Solution:
, 2
; 5 1
1
4 5 g 2
b
/ 9 .
5 9
5 cm 5 cm
x x
5 3 15 D Dg
5 4
Ch. 2 Basic Concepts of Einsteinian Relativity 57
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Using u = 3c/5 → g 55/4 and the proper length as Dx9 5 1.5 m,
2.8 The proper mean lifetime of p-mesons with a speed of 0.90c is 2.6 3 1028s. Compute their average lifetime as measured in a laboratory and the average distance they would travel before decaying.
Answer: 6.0 3 1028s, 16 m
2.9 A meterstick moves parallel to its length with a uniform speed of 0.6c, relative to an observer. Compute the length of the meter stick as measured by the observer and the time it takes for the meter stick to pass him.
Solution:
With u 5 0.6c → g 5 5/4 and Dx9 5 1 m,
2.10 Two inertial systems are separating at the uniform rate of 0.6c. If in one system a particle is observed to move parallel to the axis of relative motion between the two systems at a speed of 0.1c for 2 3 1025s, how far does the particle move, according to observers in the other system?
Answer: 480 m
2.11 How long must a satellite orbit the earth at the uniform rate of 6,711 mi/hr before its clock loses one second by comparison with an earth clock?
Solution:
With knowledge of u 5 6,711 mi/hr 5 3 3 103m/s and Dt 2 Dt9 5 1 s, we need to find Dt9. Since
Dt 2 Dt9 5 gDt9 2 Dt9 5 Dt9(g 2 1), the proper time can be expressed as
Unfortunately, since u is very small compared to the speed of light, g in the last expression is very nearly one. To avoid this difficulty, we use the result obtained in Problem 2.2,
/ . ,
and substitute into the previous expression to obtain
Using c 5 3 3 108m/s and the value for u gives Dt9 5 2 3 1010s < 634 yrs.
2.12 What must be the relative speed of separation between two inertial observers, if their time interval measurements are to differ by ten percent?
That is, for (Dt 2 Dt9)/Dt9 5 0.10, find u.
Answer: 0.417c
2.13 What must be the relative speed of separation between two inertial systems, for a length measurement to be contracted to 0.90 of its proper length?
Solution:
For this situation
Therefore, from the definition g we have
Since b 5 u/c, u = 0.436c.
2.14 A flying saucer passes a rocket ship traveling at 0.8c and the alien adjusts his clock to coincide with the rocket pilot’s watch. Twenty minutes later, according to the alien, the flying saucer passes a space station that is stationary with respect to the rocket ship. What is the distance in meters between the rocket ship and the space station, according to (a) the alien and (b) the pilot of the rocket ship?
Answer: 2.88 3 1011m, 4.80 3 1011m
2.15 Consider the situation described in Section 2.5, with the distance between A9 and B9 in S9 as 100 c-min. With u = 0.6c, compute the distance between the clocks at A9 and B9, as measured by observers in system S.
(1 ).
9 5 s 5 s
t u
c
1 2
2
1 2
2
b D
2
0.90 0.90 .
5 9 9
5 9
x x
"
gx xD D D
D
0.90 1 0.81 0.19.
2 5 2 5 5
1 b2
"
b2"
b211 , 2 1 2
.
g b
Ch. 2 Basic Concepts of Einsteinian Relativity 59
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Further, if stopwatches in S are started at the instant the flashbulb in S9 goes off, show that a stopwatch in S reads 25 min when the light flash reaches A9 and another reads 100 min when the light flash reaches B9.
Solution:
With u = 0.6c → g 5 5/4 and the proper length given as Dx9 5 100 c-min, then
2.16 According to observers in system S of Problem 2.15, how much time elapses between the activation of the two clocks in S9? How much time do they expect to have elapsed on the clock at A9 in S9, when the B9 clock is activated?
Answer: 75 min, 60 min
2.17 Two explosions, separated by a distance of 200 c-min in space, occur simultaneously to an earth observer. How much time elapses between the two explosions, according to aliens traveling at 0.8c parallel to a line con-necting the two events?
Solution:
With proper distance between the explosions being measured by the earth observer, we have Dx9 5 200 c-min and u 5 0.8c → g 5 5/3.
Thus, the aliens see the explosions occurring a distance of
apart, during the time interval
2.18 How much time will the aliens of Problem 2.17 expect to have elapsed, on an earth clock, between the occurrence of the two explosions?
Is this time interval equal to that measured by the earth observer?
/ ,
Answer: 160 min, No
2.19 Two inertial systems are uniformly separating at a speed of 0.8c. A gun fired in one system is equidistant from two observers in that system.
Both observers hear the shot 6 s after it was fired and each raises a flag. If the speed of sound in that system is 300 m/s, how much time elapses be-tween the occurrence of the two events (raising the flags), according to observers in the other system?
Solution:
In this problem we know u 5 0.8c → g 5 5/3, Dt9 5 6 s, and v9s5 300 m/s. The question is answered by finding ts, which requires that we first compute Dx9. Accordingly,
which results in
2.20 Consider the situation described for Homer and Triper in Section 2.6, with u = 0.6c and the distance between the earth and planet 9 c-yrs.
How many years will Homer and Triper age, during Triper’s voyage?
Answer: 30 yrs, 24 yrs
vs 1800 ,
9 5 9 9 5 5
s
m s m
x t 300 6
2
1D D ` j^ h
/
. .
9 5
5 9
5 3 5 3
m
m s
m s
x
c x u
c c 3600
3 5
3 10 3600 0 8
16 102
s 2 8
t g 6
D
D c
^ ^
^ ^
m h h
h h
Ch. 2 Basic Concepts of Einsteinian Relativity 61
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Introduction
The initial consideration of Einsteinian relativity in the preceding chapter was based totally on two fundamental postulates and basic physical rea-soning (logic) applied to several gedanken experiments. The results ob-tained for time dilation and length contraction were in stark contrast to the time interval (Dt 5 Dt9) and length measurement (Dx 5 Dx9) trans-formation equations predicted by Galilean relativity in Chapter 1. It was demonstrated, however, that these two relativistic effects reduced exactly
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