Question

How fast must the spaceship travel such that the clocks go twice as slow? 

Answer 1

Explanations & calculations

• according to the theory of time dilation, time runs slower for those who travel at speeds so close to the speed of light.
• Who is in motion here is the astronaut & his time $\small (\Delta \tau)$ —according to the requirement— should be half the time for an earth dweller $\small (\Delta t)$
• Then according to the time equation,

$\qquad\qquad \begin{aligned} \small \Delta t&=\small \gamma \Delta \tau\\ \small \Delta t&=\small \gamma \,(\frac{\Delta t}{2}) \cdots[\Delta \tau = 0.5\Delta t] \end{aligned}$

• Then,

$\qquad\qquad \begin{aligned} \small \gamma &=\small 2\\ \small \frac{1}{\sqrt{1-\large\frac{v^2}{c^2}}}&=\small 2\\ \small v&=\small \bold{0.866\,c} \end{aligned}$