Question

The star Arcturus is 36 light-yr from the earth (1 light-yr=9.463*10^15 m). At what speed must a space ship travel relative to the earth so that this distance is shortened to 1.0 light-yr as measured in the rest frame of the ship? What time, as measured by a clock carried in the ship, is needed for this trip from earth to start? (5%)

Answer 1

Let v be the speed of the space ship relative to earth.

And d be the Distance of star relative to the earth. d = 36 ly

Given Distance of star from Earth relative to the rest frame of ship = D = 36 ly

So,

Implies, v = (√71 /36) c

Or. v = 0.234c = 0.702 x 10⁸

Speed of ship = V = 0.702x 10⁸

Let "t" be the time required to trip to the star from Earth as measured in the ship.

t_o is the time measured in earth frame. = d/v = (36x9.463x10¹⁵)/(0.702x10⁸) = 4.85 x 10⁹ sec

sec

t = 4.72 x 10⁹ sec

t = 149.6 years