Question

A yet-to-be-built spacecraft starts from Earth moving at constant speed to the yet-tobe-discovered planet Retah, which is 20 lighthours away from Earth. It takes 25 h (according to an Earth observer) for a spacecraft to reach this planet. Assuming that the clocks are synchronized at the beginning of the journey, compare the time elapsed in the spacecraft’s frame for this one-way journey with the time elapsed as measured by an Earth-based clock.

Answer 1

Answer:

The  time elapsed at the spacecraft’s frame is less that the time elapsed at earth's  frame

Explanation:

From the question we are told that

The distance between earth and Retah is  $d = 20 \ light \ hours = 20 * 3600 * c = 72000c \ m$

Here c is the peed of light with value $c = 3.0*10^8 m/s$

The time taken to reach Retah from earth is  $t = 25 \ hours = 25 * 3600 =90000 \ sec$

The velocity of the spacecraft is mathematically evaluated  as

     $v_s = \frac{d }{t}$

substituting values

   $v_s = \frac{72000 * 3.0*10^{8} }{90000}$

    $v_s = 2.40*10^{8} \ m/s$

The time elapsed in the spacecraft’s frame is mathematically evaluated as

      $T = t * \sqrt{ 1 - \frac{v^2}{c^2} }$

substituting value

       $T = 90000 * \sqrt{ 1 - \frac{[2.4*10^{8}]^2}{[3.0*10^{8}]^2} }$

        $T = 54000 \ s$

=>    $T = 15 \ hours$

So  The  time elapsed at the spacecraft’s frame is less that the time elapsed at earth's  frame