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

Answer:

a) Bullet will hit

b) Bullets will not hit

Explanation:

Given:

The velocity of the bullet, u = $\frac{1}{3}c$ in the rest frame of the bullet pursuit car

The velocity of the original frame of reference, v = $-\frac{1}{2}c$ 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' = $\frac{1}{3}c-(-\frac{1}{2}c)$

or

u' = $\frac{1}{3}c+\frac{1}{2}c$

or

u' = $\frac{5}{6}c$

since this velocity ( $\frac{5}{6}c$) is greater than the ( $\frac{3}{4}c$)

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

$u'=\frac{u-v}{1-\frac{uv}{c^2}}$

on substituting the values we get

$u'=\frac{\frac{1}{3}c-\frac{1}{2}c}{1-\frac{\frac{1}{3}c\times \frac{1}{2}c}{c^2}}$

or

$u'=\frac{\frac{5}{6}c}{1-\frac{1}{6}}$

or

$u'=\frac{5}{7}c$

since,

$u'=\frac{5}{7}c$ is less than  ( $\frac{3}{4}c$), this means that the bullet will not hit