Question

24. As seen from Earth, two spaceships A and B are approaching along perpendicular directions. If A is observed by an Earth observer to have velocity uy ⫽ ⫺0.90c and B to have a velocity ux ⫽ ⫹0.90c, find the speed of ship A as measured by the pilot of B.

Answer 1

We take the S frame to be attached to the Earth and the S' frame to be attached to spaceship B moving with β = 0.90 along the x axis. Spaceship A has velocity components u_x = 0, u_y = −0.90c in S.

$$u_y'=\frac{u_y}{\gamma(1-\frac{u_xv}{c^2})}$$

give the velocity components of spaceship A in S' , from which 

$$u_x'=-v=-0.9c\\u_y'=\frac{u_y}{\gamma}=-0.39c\\u'=\sqrt{(-0.9c)^2+(-0.39c)^2}=0.98c$$