A5 solutions: Length contraction
1. In the time dilation problems there was a question about a rocket travelling from Earth to a star at a speed of 0.4c. The time taken for the journey measured by an observer on the Earth was 11 years. Calculate
a. The distance to the star measured from the Earth.
b. The distance between the Earth and the star measured from the rocket.
The animation below shows the signal being sent back to the Earth from the rockets frame of reference
Calculate
c. The distance travelled by the light from when it was sent to when it reached the Earth as measured by the rocket
d. The time taken for the light to reach the earth
e. On the way home, travelling at the same speed, a second signal was sent as the rocket passes the star. How many years will this signal take to get to Earth as measured by the rocket clock?
Solution
(d) In Earth's frame, the distance to the star is d_{\text{Earth}} = 4.4 \, \text{light-years}, and light travels at c. Thus, the time for the light to travel back to Earth is:
t_{\text{Earth}} = \frac{d_{\text{Earth}}}{c}
Substitute d_{\text{Earth}} = 4.4
t_{\text{Earth}} = \frac{4.4}{1} = 4.4 \, \text{years}
From the rocket’s frame, the light travels a total distance of 6.7 \, \text{light-years}, as calculated in part (c). Since light travels at c = 1 \, \text{light-year/year}:
t_{\text{rocket}} = \frac{d_{\text{light, rocket}}}{c}
Substitute d_{\text{light, rocket}} = 6.7
t_{\text{rocket}} = \frac{6.7}{1} = 6.7 \, \text{years}
(e) In the rocket's frame, the distance the light needs to travel is effectively shortened because Earth is approaching the light at v = 0.4c. Let t_{\text{rocket}} represent the time for the light to reach Earth in the rocket's frame.
From the rocket’s perspective, the total distance covered by the light is given by:
d_{\text{light}} = c t_{\text{rocket}} + v t_{\text{rocket}}
where v=0.4c
The contracted distance to Earth is d_{\text{rocket}} = 4 \, \text{light-years}. Substituting:
d_{\text{rocket}} = c t_{\text{rocket}} + v t_{\text{rocket}}
Factor out t_{\text{rocket}}:
d_{\text{rocket}} = t_{\text{rocket}} (c + v)
Substitute c = 1 \, \text{light-year/year} and v=0.4c:
4 = t_{\text{rocket}} (1 + 0.4).
Solve for t_{\text{rocket}}:
t_{\text{rocket}} = \frac{4}{1.4}. t_{\text{rocket}} \approx 2.9 years