Question

As measured in Earth's frame of reference, two planets are 489,000 km apart. A spaceship flies from one planet to the other with a constant velocity, and the clocks on the ship show that the trip lasts only 2.20 s. How fast is the ship traveling? in C

Answer 1

we know the velocity of the ship as measured by the Earth's frame must be l?/t; this is the simple velocity formula, where l? is the distance the ship travels as measured by the Earth and t is the time it takes as measured by the Earth.

Rearranging the time dilation formula we get ?=t/t? (t? is the proper time = 1.02 s). Plugging in the velocity (as defined above) into the gamma equation 1/sqrt(1-(v/c)^2), we get the following:

1/sqrt(1-(l?/(tc))^2) = t/t?

Now all it takes is a little algebra to solve for t:
t = sqrt((l?/c)^2 + (t?)^2)
t = 2.738sec
Plugging thus value for t into the velocity formula above, we get
v=(4.89*10^8 m)/(2.738 sec)
v=1.7859*10^8 m/s, or about 0.5953c <=== answer