Question

a policeman chasing an outlaw escaping on car moving at a speed of 3/4c , the police officer fires a bullet from the pursuit car which only goes half of speed of light . the muzzle velocity of bullet relative to gun is 1/3c . will bullet reach its target
according to galileo and einsteins

Answer 1

initial condition

$$v_p=\frac{1}{2}c; v_с=\frac{3}{4}c; v_b=\frac{1}{3}c$$

Galileo

velocity relative to the laboratory reference frame

$$v_b'=v_p+v_b=(\frac{1}{2}+\frac{1}{3})c=\frac{5}{6}c$$ $$\frac{5}{6}c>\frac{3}{4}c$$

The bullet will catch up with the car

Einstein

velocity relative to the laboratory reference frame

$$v_b'=\frac{v_b+v_p}{1+\frac{v_pv_b}{c^2}}=\frac{(\frac{1}{3}+\frac{1}{2})c}{1+\frac{1}{6}}=\frac{5}{7}c$$

$$\frac{5}{7}c<\frac{3}{4}c$$

The bullet will not catch up with the car