Observer O fires a light beam in the y directionVyc use thee

Observer O fires a light beam in the y direction(Vy=c) use thee lorentz velocity transformation to find V\'x and V\'y and show that O\' also measures the value of cfor speed of light.Assume that O\' moves relative to O with velocity u in the x direction

Solution

Start with T = t/sqrt(1 - (v/c)^2) the time dilation equation. Then..
cT = ct//sqrt(1 - (v/c)^2)
(cT)^2 (1 - (v/c)^2) = (ct)^2
T^2 (c^2 - v^2) = (ct)^2 and then

(cT)^2 = (ct)^2 + (vT)^2;
where cT is the distance the beam traveled as seen by O\' from outside the platform. This shows that the speed of light for the beam remains at c, but the outside observer\'s time T > t the time passage on the platform for the same event.

Note that theta = arccos(v/c) is the angle that cT travels wrt to the direction of the platform\'s travel (vT).