Question

As outlaws escape in their getaway ear, which goes 3/4 c, the police officer fires a bullet from a pursuit ear, which only goes 1/2 c. The muzzle velocity of the bullet (relative to the gun) is 1/3 c. Does the bullet reach its target according to Galileo according to Einstein?

Answer 1

a) Bullet will hit

b) Bullets will not hit

Step-by-step explanation:

Given:

The velocity of the bullet, u =
`(1)/(3)c` in the rest frame of the bullet pursuit car

The velocity of the original frame of reference, v =
`-(1)/(2)c` with respect to the pursuit car.

Now, according to the Galileo

the velocity of the bullet in the original frame of reference (u') will be

u' = u - v

on substituting the values we get

u' =
`(1)/(3)c-(-(1)/(2)c)`

or

u' =
`(1)/(3)c+(1)/(2)c`

or

u' =
`(5)/(6)c`

since this velocity (
`(5)/(6)c`) is greater than the (
`(3)/(4)c`)

hence,

the bullet will hit

Now, according to the Einstein theory

the velocity of the bullet in the original frame of reference (u') will be

`u'=(u-v)/(1-(uv)/(c^2))`

on substituting the values we get

`u'=((1)/(3)c-(1)/(2)c)/(1-((1)/(3)c* (1)/(2)c)/(c^2))`

or

`u'=((5)/(6)c)/(1-(1)/(6))`

or

`u'=(5)/(7)c`

since,

`u'=(5)/(7)c` is less than (
`(3)/(4)c`), this means that the bullet will not hit