Question

A rocket traveling at $0.50c$ sets out for the nearest star, Alpha Centauri, which is $4.3ly$ away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to

(a) stay-at-home earthlings and

(b) the rocket crew?

(c) Which answers are the correct ones, those in part a or those in part b?

Short answer

(a) Stay-at-home earthlings the total distance is $8.5ly$and time taken is $17ly$.

(b) The rockets crew's taken distance is $7.36ly$and time taken is $14.72y$.

(c) Both answers are the correct ones.

Step-by-step solution

Part(a) step 1: Given Information

We need to find how long will the journey last according to stay-at-home earthlings.

Part(a) step 2: Simplify

For the persons on the earth, the distance of the Alpha centaur is $4.25ly$, that is the total distance traveled by the voyage $4.25\times 2=8.5ly$

And the time taken earthlings is$\frac{8.5}{0,5}=17ly$

Part(b) step 1: Given Information 

We need to find out how long will the journey last according to the rocket crew.

Part(b) step2: Simplify

For the crew in spacecraft, the distance is

$l=\frac{l_0}{\gamma }=8.5\times \sqrt{1-0.5^2}=7.36ly$

time taken is

$t=\frac{t_0}{\gamma }=17\times \sqrt{1-0.5^2}=14.72y$

Part(c) step 1: Given Information 

We need to find the correct ones, those in part a or those in part b.

Part(c) step 2: Explanation

The answers in both part a and part b are correct accordingly.