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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

tvx \u394\u394 =
Use the time-dilation equation to
relate the mean lifetimes of the pions
in the accelerator laboratory \u394t to

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.
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.
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```