We all know about the ultimate speed limit... the speed of light.

If person A stands on Earth and shoots his pistol, he observes the bullet to fly directly away at 1000 mph. Person B is standing right next to him (not in front) and watches this event, and agrees that the bullet flies directly away at 1000 mph.

Let's change the situation and say that B is in a spaceship, and A is in a different (and very long) spaceship with lots of windows. B's ship is hovering in space (no thrusters/acceleration). A's ship is approaching from a distance and is going to pass B's ship (very close) but at incredible speed. Make careful note that A's ship is NOT thrusting or accelerating at all, it is "coasting". In fact, A's ship is moving, relative to B's ship at 10 mph less than the speed of light. WOW!

A stands in the middle of his ship and points his gun directly forward (in the direction of travel), and fires the same pistol at the exact moment that he is passing B.

The questions are: How fast does he observe the bullet leave the gun? How fast does B observe the bullet leave the gun?

How do your answers change (if at all) if A aims backwards when he fires?

(In reply to solution by Charlie)

Although I used a different framing story (based on the "twin paradox"), I have been working on this puzzle off and on for about 30 years. I have gotten as far as Charlie's answer:

        s1 + s2        
  1 + [(s1)(s2)/c²]

That is as far as the basics of Special Relativity could take me.

My particular framing story did make it clear, however, that there is a velocity difference that is conceptually related to the Doppler shift effect. Since Relativity has not yet been reconciled with Quantum Mechanics, I had harbored hopes that showing that particles can obey what is considered to be a rule for waves I could open the way toward such a reconciliation. But alas, I have never been able to translate the Doppler shift into a velocity difference for particles travelling at less than the speed of light.*

BTW, SK, the math would have been a lot easier if you'd expressed the speeds as fractions of c (for example, the ship travelling at .95c and the bullet at .01c relative to the ship). Or was that deliberate, to make the problem "trickier"?

*I even tried considering a "meta-velocity" of which the velocity which is measured in Special Relativity is the hyperbolic tangent, since tanh(a + b) = [tanh(a) + tanh(b)]/[1 + tanh(a)tanh(b)], but it didn't help.

Posted by TomM on 2003-11-29 03:24:58