Question

A rocket is on its way to Mars to explore the possibility of life on the planet.

1. How fast must a rocket travel relative to the Earth so that time in the rocket “slows down” to half its rate as measured by Earth-based observers?

2. Do present-day jet planes approach such speeds? This is not simply a yes or no answer.

Answer 1

Let the speed be v
Let the time taken in rocket be t
Time as observed by earth based observers = t'
Given,
t = t'/2

Due to time dilation concept,


t' = t / sqrt(1 - v^2/c^2)
2t = t /sqrt(1 - v^2/c^2)
sqrt(1 - v^2/c^2) = 1/2
(1 - v^2/c^2) = 1/4
1 - 1/4 = v^2/c^2
v = sqrt(3/4) * c
v = 0.866 c

Where c is the speed of light, = 3.0 * 10^8 m/s

v = 2.6 * 10^8 m/s

(2)
No !!
The speed we need is too high !!, Max speed present-day jet planes approach is apprximately 1017 m/s