# Does relativity restrict velocity relative to an observer?

1. Jul 26, 2012

### jfraze

I know this is a stupid question but I can't figure out how to get an answer. I'm trying to figure out if the rules of relativity force a limit on the velocities of two objects in relation to each other. Just as a thought experiment, if a spaceship is traveling at 186,281 miles per second over a finite distance from point A to point B, and fires a missile at 1 mile per second relative to itself, that missile should be traveling at 186,282 miles per second relative to points A and B, which is not possible. So would relativity actually prevent the hypothetical missile from traveling at 1 mile per second relative to the hypothetical spaceship? (To take it further, what if the spaceship were traveling 186,281.999999 miles per second... the fastest it could fire a missile, relative to itself, would have to be less than .000001 miles per second. It seems observers on the spaceship would see that missile barely creeping away from them, even relative to their own spacetime.) My best guess on a solution is that since time is compressed for the observers on the spaceship, the missile might be able to travel at 1 mile per second according their experience of what a "second" is, but would still be less than the speed of light relative to points A and B. I may be off on the wrong track with that, though.

OK, you can commence mocking me now.

2. Jul 26, 2012

### oktovan

The missile won't travel faster than the speed of light. Instead of gaining speed, it will gain mass.

3. Jul 26, 2012

### Staff: Mentor

That's pretty much right, although it's not just time dilation that makes the speeds cpme out that way; length contraction and relativity of simultaneity also contribute.

Spaceship fires a missile that travels at speed u relative to the ship. The ship is moving at speed v relative to some observer. The speed of the missile relative to the observer is not u+v as classical mechanics and our experience with objects moving at speeds much less than the speed of light suggests. It will be $$\frac{u+v}{1+\frac{uv}{c^{2}}}$$
which is never greater than c, and is indistinguishable from u+v for the speeds that we're familiar with day to day.