Homework 7 solutions
Physics 204 evening summer 2008
1.
REASONING
The distance between earth and the center of the galaxy is the proper length
L
0
, because
it is the distance measured by an observer who is at rest relative to the earth and the center of the galaxy.
A
person on board the spaceship is moving with respect to them and measures a contracted length
L
that is related
to the proper length by
()
22
0
1/
LL
vc
=−
.
The contracted distance is also equal to the product of the
spaceship’s speed
v
the time interval measured by a person on board the spaceship.
This time interval is the
proper time interval
Δ
t
0
because the person on board the spaceship measures the beginning and ending events
(the times when the trip starts and ends) at the same location relative to a coordinate system fixed to the
spaceship.
Thus, the contracted distance is also
L
=
v
Δ
t
0
.
By setting the two expressions for
L
equal to each other, we can find the how long the trip will take
according to a clock on board the spaceship.
SOLUTION
Setting
0
equal to
L
=
v
Δ
t
0
and solving for the proper time interval
Δ
t
0
gives
0
0
15
2
27
8
9.47 10
m
23 500 ly
1ly
0.9970
1yr
1
1822yr
3.16 10 s
0.9970 3.00 10 m/s
Δ=
−
⎛⎞
×
⎜⎟
⎡⎤
⎝⎠
⎢⎥
×
×
⎣⎦
L
tv
c
v
c
c
=
2.
REASONING
Assume that traveler A moves at a speed of
v
A
= 0.59
c
and traveler B moves at a speed of
v
B
= 0.83
c
, both speeds being with respect to the earth.
Each traveler is moving with respect to the earth
and the distant star, so each measures a contracted length
L
A
or
L
B
for the distance traveled.
However, an
observer on earth is at rest with respect to the earth and the distant star (which is assumed to be stationary
with respect to the earth), so he or she would measure the proper length
L
0
.
For each traveler the
contracted length is given by the lengthcontraction equation as stated in Equation 28.2:
AB
A0
B0
1
and
1
vv
cc
It is important to note that the proper length
L
0
is the same in each application of the lengthcontraction
equation.
Thus, we can combine the two equations and eliminate it.
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 Spring '09
 rollino
 Physics, Energy, Kinetic Energy, Special Relativity, lb

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