Q:

How did they prove light does not move faster on a moving object?

I always see that the speed of light is a constant and never changes. But has this ever been measured? 
Have they taken a stationary object and measured the speed of light coming from it in a vacuum and then also measured the speed of light coming from a moving object in the same direction as the direction of the light? 
This may be a silly question, but we do have the tools to measure it now. 

A:

Yes, the experiment was famously done by Michelson and Morley in 1887.
They used the Earth itself as the moving object, and measured the speed of light in the east-west direction (in the direction the Earth is moving) and compared it with the speed of light in the north-south direction (in which the Earth is roughly stationary, at least relative to the Sun). Others later repeated the experiment at different times of year, when the Earth's velocity is in different directions relative to the path of the Sun through the galaxy.

