special relativity - Something almost faster than light traveling on something else almost faster than light

Possible Duplicate:
Travelling faster than the speed of light
Double light speed

Lets say that

• an airplane can fly at 4/5 of the speed of light, and


• I can run at 2/5 of the speed of light, and


• I'm in the airplane.

Suddenly I start running towards the cockpit down the middle lane. Now what?

From what I know, the speed of light is not the speed of light just because it is. The speed of light is instead the maximum speed limit because light moves as fast as possible and therefore equals the max speed.
But when it's the maximum speed limit, nothing can ever move faster.

So, I'm running in the airplane... now what?

Answer

This must have been asked before, but the only near duplicate I can find is How to derive addition of velocities without the Lorentz transformation? and this isn't an exact duplicate.

The point is that relativistic velocities can't just be added. In your example let u be the plane's speed, 0.8c, and v be your speed, 0.4c, then the speed a stationary observer sees, w, is given by:

$$ w~=~\dfrac{u+v}{1+uv/c^2} $$

which is about 0.91c.

The other passengers in the plane will see you running at 0.4c, but remember that the aeroplane's time is running more slowly than a stationary observers time. So what looks like 0.4c to the passengers on the plane looks slower to me standing still on the ground.