Question

You have a spacecraft that is capable of reaching a speed of 0.99c and desire to travel to Vega, a nearby star about 25 light years away. In order to keep track of your position, your spacecraft sends out a flash of light every second (by your ship’s clock) during the journey that can be picked up on Earth. In addition, the Earth sends out a flash every second (by the Earth clock) that you can see with a telescope on your ship.

Thus equipped, you depart to your destination, where after arrival you send a radio signal to Earth saying that you have arrived. After spending 0.5 years there, you send a second radio signal that says you are departing to return to Earth.

In answering the following questions, Ignore the negligible time it takes to accelerate and decelerate your spacecraft, and assume Earth and Alpha Centauri have negligible relative motion (compared to the speed of light!):

(a) On the outward journey, what is the frequency of flashes seen coming from your spacecraft back on Earth? Similarly, what is the frequency of flashes you see coming from Earth?

(b) After getting up to 0.99c, how far away does Vega appear to you at the start of the journey? How many years does it take to get there according to your spacecraft clock?

(c) How many years (on the Earth’s clock) after you left Earth will they receive the radio signal that you have arrived?

(d) While you are at Vega, what is the frequency of the flashes the Earth sees from your spacecraft? What is the frequency you see from the Earth’s flasher?

(e) What is the frequency of the Earth’s flasher you see on the way back? What is the frequency of your flasher as seen by the Earth?

(f) How many years after you left (by the Earth’s clock) will the Earth receive the signal that you have started back?

(g) How many years after you left (by the Earth’s clock) will have elapsed when you get back? How much older will you be?

Answer 1

a) speed of the space craft v = 0.99c

= 1/sqrt(1-v²/c² ) = 7.09

time as per moving frame t' = 1s

time as per rest frame (Earth) = t' = 7.09 s

however the light from the space craft has to travel an additional distance of 7.09*0.99c as the space craft is receding from earth

Total time inetervel between flashes = 7.09 s + 7.09*0.99c /c = 14.11s

As seen from the space craft earth is receding from it.

hence = 7.09

The space craft will see the light flahses from earth at the same frequence i.e. 14.11s /flash

b) distane of Vegas = 25 Ly

the distance measured is from Earth which is at rest.

the distance will contract as seen from the space craft which is movign at a speed of 0.99s

length measured wrt space craft = L_p / = 25/7.09 = 3.526 Ly

c) Time to reach vegas as measured from the space craft = 3.526/0.99 = 3.526 Yr

time as seen from earth = 3.526*7.09 = 25.252 Yrs

Earth will recive the flash from vegas after 25.252 Yrs of leaving the earth.

d) When at vegas bothe are at rest and there is no change in the time are distance measurment. Both frames are identical.

both will recieve flashes at 1s intervel.

e) on return journey releative speeds are same but the light source is approaching the observer and it has to travel less distance between flash intervels

hence time intervel between flashes = 7.09 - 7.09*0.99 = 0.0709 s

I is similar to relativistic doppler effect.

f) In earth frame the space craft reached vegas after 25.252 Yrs it has left earth. It stayed there for 0.5 Yrs

It starts bakc after 25.752 yrs after it left earth.

g)When the space craft reachd back earth total time elapsed

= 25.252*2 +0.5 = 51.004 Yrs elapsed. from the time it has left.