ChR ISM
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ChR ISM


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2
2
0
1
\u394
\u394
c
v
tt
\u2212
= \u21d2
2
01 \u239f\u23a0
\u239e\u239c\u239d
\u239b
\u394
\u394\u2212=
t
tcv 
Substitute numerical values and evaluate v: 
 
( ) m/s108.2
s105.7
s106.21m/s10998.2 8
2
8
8
8 ×=\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b
×
×\u2212×= \u2212
\u2212
v 
 
23 \u2022 A meterstick moves with speed 0.80c relative to you in the direction 
parallel to the stick. (a) Find the length of the stick as measured by you. (b) How 
long does it take for the stick to pass you? 
 
Picture the Problem We can find the measured length L of the meterstick using ( )220 1 cvLL \u2212= and the time it takes to pass you using L = v\u394t. 
 
(a) Express the length L of the 
meterstick in terms of its proper 
length L0: 
 
2
2
0 1 c
vLL \u2212= 
Substitute numerical values and 
evaluate L: ( ) ( ) cm608.01m0.1 2
2
=\u2212=
c
cL 
Chapter R 
 
 
1054 
(b) Express the time it takes for the 
meterstick to pass you in terms of its 
apparent length and speed: 
 
v
Lt =\u394 
Substitute numerical values and 
evaluate \u394t: ns5.28.0
m60.0
\u394 ==
c
t 
 
24 \u2022 Recall that the half-life is the time it takes for any given amount of 
unstable particles to decay to half that amount of particles. The proper half-life of 
a species of charged subatomic particles called pions is 1.80 × 10\u20138 s (See 
Problem 18 for details on pions.) . Suppose a group of these pions are produced 
in an accelerator and emerge with a speed of 0.998c. How far do these particles 
travel in the accelerator\u2019s laboratory before half of them have decayed? 
 
Picture the Problem We can express the distance the pions will travel in the 
laboratory using tvx \u394\u394 = and find their half-life in the accelerator laboratory 
using ( )20 1\u394\u394 cvtt \u2212= . 
 
Express the average distance the 
pions will travel before decaying in 
terms of their speed proper mean 
lifetime: 
 
tvx \u394\u394 = 
Use the time-dilation equation to 
relate the mean lifetimes of the pions 
in the accelerator laboratory \u394t to 
their proper mean lifetime \u394t0: 
 
2
0
1
\u394
\u394
\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212
=
c
v
tt 
Substitute for t\u394 to obtain: 
 2
0
1
\u394
\u394
\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212
=
c
v
tvx 
 
Substitute numerical values and 
evaluate \u394x: 
( )( ) m2.85
998.01
s1080.1998.0
\u394
2
8
=
\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212
×=
\u2212
c
c
cx 
 
25 \u2022\u2022 [SSM] Your friend, who is the same age as you, travels to the star 
Alpha Centauri, which is 4.0 light-years away, and returns immediately. He 
claims that the entire trip took just 6.0 y. What was his speed? Ignore any 
accelerations of your friend\u2019s spaceship and assume the spaceship traveled at the 
same speed during the entire trip. 
Special Relativity 
 
 
1055
Picture the Problem To calculate the speed in the reference frame of the friend, 
who is named Ed, we consider each leg of the trip separately. Consider an 
imaginary stick extending from Earth to Alpha Centauri that is at rest relative to 
Earth. In Ed\u2019s frame the length of the stick, and thus the distance between Earth 
and Alpha Centauri, is shortened in accord with the length contraction formula. 
As three years pass on Ed\u2019s watch Alpha Centauri travels at speed v from its 
initial location to him. 
 
Sketch the situation as it is in your 
reference frame. The distance 
between Earth and Alpha C. is the 
rest length L0 of the stick discussed 
in Picture the Problem: 
 
Alpha C.
Your reference frame
v
Earth
Ed moving
0L
 
 
Sketch the situation as it is in Ed\u2019s 
reference frame. The distance 
between Earth and Alpha C. is the 
length L of the moving stick 
discussed in Picture the Problem: 
 
Alpha C.
Your friend Ed's reference frame
v
Earth
Ed at rest
L
v
 
 
The two events are Ed leaves Earth 
and Ed arrive at Alpha Centauri. In 
Ed\u2019s frame these two events occur at 
the same place (next to Ed). Thus, 
the time between those two events 
\u394t0 is the proper time between the 
two events. 
 
\u394t0 = 3 y 
 
The distance L traveled by Alpha 
Centauri in Ed\u2019s frame during the 
first three years equals the speed 
multiplied by the time in Ed\u2019s frame: 
 
0L v t= \u394 
 
 
The distance between Earth and 
Alpha Centauri in Ed\u2019s frame is the 
contracted length of the imaginary 
stick: 
 
2
0 21
vL L
c
= \u2212 
Equate these expressions for L to 
obtain: 
 
2
0 021
vL v t
c
\u2212 = \u394 
 
Chapter R 
 
 
1056 
Substituting numerical values yields: ( ) ( )y 0.31y 0.4 2
2
v
c
vc =\u2212\u22c5 
or 
c
v
c
v 75.01 2
2
=\u2212 
 
Solve the quadratic equation in v/c ot 
obtain: 80.0=c
v \u21d2 cv 80.0= 
 
26 \u2022\u2022 Two spaceships pass each other traveling in opposite directions. A 
passenger in ship A knows that her ship is 100 m long. She notes that ship B is 
moving with a speed of 0.92c relative to A and that the length of B is 36 m. What 
are the lengths of the two spaceships as measured by a passenger in ship B? 
 
Picture the Problem We can use the relationship between the measured length L 
of the spaceships and their proper lengths L0 to find the lengths of the two 
spaceships as measured by a passenger in ship B. 
 
Relate the measured length LA of 
ship A to its proper length: 
 
2
A,0A 1 \u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212=
c
vLL 
 
Substitute numerical values and 
evaluate LA: 
 
( ) m3992.01m100 2A =\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212=
c
cL 
 
Relate the proper length L0,B of ship 
B to its measured length LB: 
 
2
B
B,0
1 \u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212
=
c
v
LL 
 
Substitute numerical values and 
evaluate B,0L : 
m92
92.01
m36
2B,0
=
\u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212
=
c
c
L 
 
27 \u2022\u2022 Supersonic jets achieve maximum speeds of about 3.00 × 10\u20136c. (a) By 
what percentage would a jet traveling at this speed contract in length? (b) During 
a time of exactly one year or 3.15 × 107 s on your clock, how much time would 
elapse on the pilot\u2019s clock? How much time is lost by the pilot\u2019s clock in one year 
of your time? Assume you are on the ground and the pilot is flying at the specified 
speed for the entire year. 
 
Picture the Problem We can use the relationship between the measured length L 
of the jet and its proper length to express the fractional change in the length of the 
Special Relativity 
 
 
1057
jet traveling at its maximum speed. In Part (b) we can express the elapsed time on 
the pilot\u2019s clock \u394t0 in terms of the elapsed time \u394t on your clock. 
 
(a) Express the fractional change in 
length of the jet: 
 
00
0 1
L
L
L
LL \u2212=\u2212 
Relate L to L0: 2
0 1 \u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212=
c
vLL \u21d2
2
0
1 \u239f\u23a0
\u239e\u239c\u239d
\u239b\u2212=
c
v
L
L 
 
Substitute for 0LL to obtain: 
 2
2
0
0 11
c
v
L
LL \u2212\u2212=\u2212 (1) 
 
Expand 2
2
1
c
v\u2212 binomially to 
obtain: 
 
2
2
4
4
2
2
2
2
2
11
...
8
3
2
111
2
1
c
v
c
v
c
v
c
v
\u2212\u2248
++\u2212=\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212
 
 
Substituting in equation (1) yields: 
2
2
2
2
0
0
2
1
2
111
c
v
c
v
L
LL =\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212\u2212=\u2212 
 
Substitute numerical values and 
evaluate the jet\u2019s fractional change in 
length: 
 
( )
%1050.4
1000.3
2
1
10
2
26
0
0
\u2212
\u2212
×=
×\u2248\u2212
c
c
L
LL
 
 
(b) Express the elapsed time on the 
pilot\u2019s clock \u394t0 in terms of the 
elapsed time \u394t on your clock: 
 t
c
vt
c
vtt
c
vt
\u394
2
1
\u394
2
11\u394\u3941\u394
2
2
2
2
2
2
0
2
1
\u2212=
\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212\u2248\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212=
 
where the second term represents the 
time lost on the pilot\u2019s clock. 
 
Substitute numerical values and evaluate the elapsed time on the pilot\u2019s clock 
in 1 y = 3.15 × 107 s: 
 ( ) ( ) y 00.1s1015.31000.3
2
11\u394
2
11\u394 72
26
2
2
0 =×\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b ×\u2212=\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b \u2212=
\u2212
c
ct
c
vt 
 
Chapter R 
 
 
1058 
Substitute numerical values and 
evaluate the time lost on the pilot\u2019s 
clock in 1 y = 3.15 × 107 s: 
 
( ) ( )
s 421
s1015.31000.3
2
1
\u394
2
1 7
2
26
2
2
\u3bc=
××=
\u2212
c
ct
c
v
 
 
28 \u2022\u2022 The proper mean lifetime of a muon (see Problems 20 and 21 for 
details) is 2.20 \u3bcs. Consider a muon, created in the Earth\u2019s atmosphere, speeding 
toward the surface 8.00 km below, at a speed of 0.980c. (a) What is the 
likelihood that it will survive its trip to ground before decaying? The probability 
of a muon decaying is given by P = 1\u2212 e\u2212\u394t /\u3c4 , where \u394t is the time interval as 
measured in the reference frame in question. (b) Calculate this from the point of 
view of an observer moving with the muon. Show that, from the point of view of