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Relativistic bullet (Posted on 2003-11-28) Difficulty: 5 of 5
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?

See The Solution Submitted by SilverKnight    
Rating: 3.5000 (8 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 4 of 20 |
The formula for the speed of an object when that object is moving at speed s1 with respect to another object that is already moving at speed s2 in the same direction is (s1+s2)/(1+s1*s2/c^2), where c is the speed of light.

The speed of light is 670616629.384395 mi/hr, so A's ship is moving at 670616619.384395 mi/hr relative to B's.

So the bullet, relative to B is travelling at

(1000+670616619.384395) / (1+1000*670616619.384395/(670616629.384395^2))

= 670616619.384424 mi/hr, or just .000029 mi/hr more than A's ship itself.

The alternative version, with A shooting rearward, just reverses the sign of the 1000 mi/hr:

(-1000+670616619.384395) / (1-1000*670616619.384395/(670616629.384395^2))

= 670616619.384365 mi/hr, or just .000030 mi/hr less than A's ship itself.

Of course, relative to A, the bullet travels at 1000 mi/hr, regardless of the direction.

  Posted by Charlie on 2003-11-28 19:37:18
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