Question

If a spaceship of proper length 40 m is measured to have a length 30 m, how fast is it moving?

Answer 1

In special relativity, the length contraction of an object moving with speed v is given by

`L=L_0 \sqrt{1- (v^2)/(c^2) }`
where
L is the measured length
L0 is the proper length
v is the speed of the object
c is the speed of light

In our problem, the measured length is 30 m while the proper length was 40 m, therefore if we re-arrange the equation we can calculate the speed of the spaceship:

`v=c \sqrt{1- ((L)/(L_0))^2 }=(3 \cdot 10^8 m/s ) \sqrt{1-( (30m)/(40 m) )^2}=1.98 \cdot 10^8 m/s`