Suppose you're traveling on a space ship at 9/10 the speed of light (.9c). You have a high-powered rifle that shoots bullets at the same speed. Suppose you shoot the bullet perpendicular to your direction of travel.

It appears that the bullet would travel at a 45-degree angle (northeast, if the ship is traveling north and the bullet is shot eastward), at about 1.2728c which is faster than light. Why is this wrong, and what would the actual speed and direction be?

Why might θ not be 45°?

A distance parallel to the direction of travel measured by a ruler in the vehicle will be √[1-(v²/c²)] shorter than than the same distance measured by a ruler on the planet. (Lorentz contraction). A distance perpendicular to the direction of travel measured by the two rulers will be the same.

So θ is not arctan 1/1 = 45° but arctan (1/[1-(v²/c²)]) = arctan (1/[1-.81]) = arctan (1/.19) =arctan (5.2632)

(Also the in the formula I supposed earlier tan θ/[1+v(V)v(M)/c²], the function is more likely to be cos than tan.)

If θ = arctan (5.2632), the Newtonian vector sum would give a speed of √[.9c² + .19c²]=√[.81c² + 0.0361c²]= √[.8461c²]=.9198c

Applying my (revised) speed reduction formula yields (cos[arctan(5.2632)])/[1+(.9)(.0361)]

If I were more sure of my formulae, the completion of this problem would now be a simple handwaving exercize.

Posted by TomM on 2005-01-18 08:45:42