A getaway car traveling at 3/4c is pursued by a police car traveling at 1/2c. The police car fires a bullet which travels at 1/3c (relative to the gun) towards the getaway car. Does the bullet reach the car
a) according to Galileo’s theory b) according to Einstein’s theory?
a) Speed of bullet relative to the ground is 1/2c + 1/3c = 5/6c = 10/12c
Speed of getaway car = 9/12c. So bullet reaches the car.
b) Speed of bullet relative ground = (1/2c + 1/3c)/(1 + ½*1/3) = 20/28c
Speed of getaway car = 3/4c = 21/28c. So bullet doesn’t reach car.
But what does the police car see? It would seem that it would see the getaway car traveling at 3/4c - 1/2c = 1/4c and the bullet traveling at 1/3c and would therefore see the bullet reach the getaway car. But this can’t
be right so what am doing wrong?
I don’t know the answer to this question and I don’t know if what I’m about to say even has any bearing on it but FWIW adding relativistic velocities doesn’t work the way you add velocities in the ‘normal’ world we are used to.
For instance, I am travelling at .9c and fire a gun that moves the projectile out at .9c. The projectile does not move at 1.8c (impossible). Here on earth if I am in a plane moving 500 mph and fire a gun inside that has a bullet velocity of 500 mph the bullet is moving at 1000 mph.
Strange stuff.
(Your problem seems more like a mathematical trick…like the missing dollar when the bell boy refunds less money than he should problem.)
I should note that in my example the bullet in the plane is moving at 1000 mph relative to the ground…the bullet is moving 500 mph relative to the plane.
Whack-a-Mole I used the relativistic velocity addition formula in part (b) of my problem and it shows that the bullet will never reach the getaway car.
But when I go to the police car frame it would seem that the driver would see the getaway car receding at 1/4c and the bullet receding at 1/3c and the bullet would therefore overtake the getaway car. Can this be right?
I’m not sure what you did in equation B. In both cases you are comparing the speed of the items relative to the ground in which case your answer is in A. The bullet catches the getaway car.
Try it this way and assume ALL speeds are measured by an observer who is not moving (the ground in your example):
Get away car – .75c
Police car – .5c
Bullet – .33c
Police fire bullet and velocities add --> .5 + .33 - .83c
Oops…my bad. I see what you are getting at in B (from your second post although I can only assume you applied the relativistic formula necessary for doing this).
In this case go with Einstein and adding relativistic speeds. As I said before velocity doesn’t add as one would guess from our day-to-day experiences. At relativistic speeds time dilation becomes an issue and I think that is what starts to gum up the works.
What you’re doing wrong is forgetting that those velocities are relativistic. The getaway car is going 3/4c relative to the ground, but (3/4-1/2)/(1-3/4*1/2)c=2/5c relative to the police, faster than the bullet.
What you’re doing wrong is forgetting that those velocities are relativistic. The getaway car is going 3/4c relative to the ground, but (3/4-1/2)/(1-3/4*1/2)c=2/5c relative to the police, faster than the bullet.