Moving Faster Than the Speed of Light? Exploring the Possibilities

[Pineapple]

I know that according to e=mc^2, it should be impossible to accelerate anything with mass to the speed of light, but what if someone is on a large spaceship moving at 80% of c, and are themselves moving at, say 20 or 30% of the speed of light within the spaceship.

According to an immobile observer, outside the spaceship wouldn't this person be moving faster than the speed of light?

Of course I know this must be wrong, but I'm wondering what the flaw in my reasoning is.

 

[Doc Al]

Your question isn't naive. Your error is thinking that speeds add like this:

$$V_{a/c} = V_{a/b} + V_{b/c}$$

When they really add like this:

$$V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}$$

The first (called Galilean addition of velocities) is only an approximation good for low speeds; the second is the more accurate relativistic addition of velocities. You'll see that speeds will never add to more than c.

 

[Frame Dragger]

I know that according to e=mc^2, it should be impossible to accelerate anything with mass to the speed of light, but what if someone is on a large spaceship moving at 80% of c, and are themselves moving at, say 20 or 30% of the speed of light within the spaceship.

According to an immobile observer, outside the spaceship wouldn't this person be moving faster than the speed of light?

Of course I know this must be wrong, but I'm wondering what the flaw in my reasoning is.

This is sort of a reformulation of the classic "light flash" thought experments. No matter how you slice it, the speed of light in a given medium is STATIC. I would reach about the basics of Special Relativity, which basically takes the conundrum you're struggling with, and solves it.

This is WHY things are relative, because c is invariant. If c is the same everywhere, and if all observers have to "agree on the laws of physics", then it's SPACETIME which varies. This is how you get to Relativity in the first place.

If you hold a flashlight, and walk forward... the velocity of your motion is not added to the velocity of the light. The interval between the light reaching X target is a function of the medium and distance (geodesic followed).

EDIt: Doc Al beat me to, and probably for the best. Sorry Doc!