Question

The diameter of the solar system is 10 light hours. A spaceship crosses the solar system in 15 hours, as measured on earth. How long, in hours, does the passage take according to passengers on the spaceship?

Short answer

According to passenger on spaceship it will take 11.2 h.

Step-by-step solution

Given information 

We have given that, the diameter of the solar system is 10 light hours. A spaceship crosses the solar system in 15 hours, as measured on earth.

Step 1.

We assume that earth is S frame of reference and spaceship be S' frame of reference. S' moves relative to S with speed v.

For an observer on earth, the length of the solar system is 10 lh.

The time interval for the spaceship to cross is $∆t=15hours$.

The time measured in S' frame is the proper time because this can be measured with one clock at both position.

Step 2.

The velocity is :

$v=\frac{10lh}{15h}$

$v=\frac{2}{3}c$

Because $\Delta t\text{'}=\Delta \zeta$

$\Delta \zeta =\Delta t\sqrt{1-{\beta }^2}$

$\Delta \zeta =(15h)\sqrt{1-{\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^2}$

$\Delta \zeta =11.2h$