PC 242 Assignment 1 Solutions
1. An athlete runs the 100 meter dash at 10 m/s. How much will her watch gain or lose as
compared to ground based clocks during the race?
According to Einstein’s theory of special relativity, moving clocks tick slower than
stationary clocks, so the athlete’s watch will lose time.
The proper time taken by the athlete to run the 100m in the stationary frame of the earth
100m
is !t =
= 10s
10m / s
The time dilation in the athlete’s frame is
1
v2
100 2
!t " = !t = 1 $ 2 !t = 1 $
2 10s & 10s
#
c
3 % 10 8
(
)
The athlete is running too slowly relative to the speed of light to measure any discernible
time dilation.
2. When he returns his Hertz rent-a-rocket after 1 week’s cruising in the galaxy, Mr.
Spock is shocked to find that he has been billed for a 3 week rental. Assuming that he
traveled straight out and straight back always at constant speed, how fast was he going?
Mr. Spock is apparently unaware of Einstein’s theory of special relativity, which states
that moving clocks tick slower than stationary clocks.
In the earth stationary frame, the time elapsed is !t = 3weeks .
In the frame of the rocket, the time elapsed is !t " = 1week .
According to the time dilation formula,
1
1
v 2 !t " 2 1
!t " = !t $ 2 = 1 % 2 = 2 =
#
#
c
!t
9
v2 8
= $ v = 0.94c
c2 9
Mr. Spock was speeding around at close to the speed of light and hence experienced
significant time dilation.
3. A relativistic conveyor belt is moving at speed 0.5c relative to a stationary frame S.
Two observers standing beside the belt 10 metres apart as measured in S paint a mark on
the belt at the same instant as measured in S. How far apart will the marks be as
measured by observers on the conveyor belt?
Observers on the conveyor belt are stationary in the frame of the conveyor belt. Relative
to this frame observers beside the belt are moving at v=-0.5c.
According to Einstein’s theory of special relativity, the distance L=10 metres between the
observers standing beside the belt will be contracted to
1
v2
L ! = L = 1 # 2 L = 1 # 0.25L = 0.87 $ 10m = 8.7m
"
c
The distance between the marks is measured by observers on the belt to be 8.7m.
4. How fast and in what direction must galaxy A be moving if an absorption line found at
550nm (green) for a stationary galaxy is shifted to 700nm (red) for A?
As a consequence of the special theory of relativity, a red shift occurs when a source is
moving away from an observer, so galaxy A is moving away from us. The relativistic
Doppler shift is
1! v c
fobs =
fsource
1+ v c
Wavelength is related to frequency by f = c / !. Thus
!source
1" #
=
!observer
1+ #
solve for # =
v
c
2
2
2
+$ !
.
$ !source '
$ !source '
'
source
1
+
#
=
1
"
#
*
"
1
=
"
#
+
1
0
(
)
&% !
)
&% !
)
&
)
-,% !observer (
0/
observer (
observer (
# =
$ !
'
1 " & source )
% !observer (
2
$ !
'
1 + & source )
% !observer (
2
#=
=
1 " ( 550 700 )
1 + ( 550 700 )
2
2
=
0.383
1.617
v
= 0.24 * v = 7.1 1 10 7 m/s
c
5. Two spaceships approach each other moving with the same speed as measured by an
observer on Earth. If their relative speed is 0.70c, what is the speed of each spaceship?
S: rest frame of the Earth
S’: frame of the spaceship moving at speed v to the right relative to Earth.
2nd spaceship moving to the left with speed u = -v relative to the earth.
Speed u’ of 2nd spaceship in the S’ reference frame is 0.70c
Using the Lorentz transformations to relate the speed of the second ship in the two
frames,
ux ! = 0.70c =
ux " v
1" vux c
2
=
"v " v
1+ v
2
c
2
=
"2v
1+ v 2 c 2
# v2 &
v2
v
0.70c % 1+ ( = "2v ) 1+
= "2.86
%$ c 2 ('
c
c2
Setting
v
= *,
c
* 2 + 2.86* + 1 = 0 ) * =
v
= "0.41 ) v = "0.41c
c
The speed of each spaceship is 0.41c. The velocities are 0.41c and -0.41c.
6. A rocket traveling at 0.8c sends out a beam of particles which travel at 0.9c relative to
the rocket. What is the particles’ speed relative to earth ? If the rocket shoots particles
traveling at speed c, what is the speed of the particles relative to earth?
S: rest frame of the Earth
S’: frame of the spaceship moving at speed v=0.8c to the right relative to Earth.
Speed u’ of particles in the S’ reference frame is 0.9c
Using the Lorentz transformations to relate the speed of the particles in the two frames,
ux =
ux ! + v
1+ vux ! c
2
=
0.9c + 0.8c
1+ (0.8)(0.9)c
2
c
2
=
1.7c
= 0.99c
1.72
The particles move with speed 0.99c relative to the Earth. Their velocity is 0.99c in the
same direction as the rocket.
If the particles are traveling at c relative to the rocket then in the Earth’s frame
ux =
ux ! + v
1+ vux ! c
2
=
c + 0.8c
1+ (0.8)c
2
c
2
=
1.8c
=c
1.8
The speed of the particles is c in both frames. The particles travel at the speed of light –
they must be photons. This confirms Einstein’s statement that the speed of light is
constant in all frames.