Question

At what speed, as a fraction of c, must a rocket travel on a journey to and from a distant star so that the astronauts age 15 years while the Mission Control workers on earth age 130 years?

Answer 1

Answer:

The velocity as a fraction of c is 0.986 c m/s

Solution:

As per the question:

Time measured by the astronaut, t = 15 yrs

Time measured in the frame of mission control, t' = 130 yrs

Now,

Using the formula of time dilation:

$t' = \frac{t}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Substituting appropriate values in the above eqn:

$130 = \frac{15}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

$\sqrt{1 - \frac{v^{2}}{c^{2}}}= \frac{15}{130}$

Squaring both the sides we get:

$1 - \frac{v^{2}}{c^{2}}= (\frac{15}{130})^{2}$

$\frac{v^{2}}{c^{2}} = 1 - (\frac{15}{130})^{2}$

v = 0.986 c m/s