The outlaws are escaping in their getaway car, that goes ${3\over4}c$, chased by the police, moving at only ${1\over2}c$. Realizing they can't catch up, the police attempt to shoot out the tires of the getaway car. Their guns have a muzzle velocity (speed of the bullets relative to the gun) of ${1\over3}c$.

1. Does the bullet reach its target according to Galileo?
2. Does the bullet reach its target according to Einstein?
3. Verify that your answer to question 2 is the same in all four (!) reference frames: ground, police, outlaws, and bullet.

1. In Newtonian physics, we add the speed of the bullet ($\frac12c$) to that of the police ($\frac13c$) to compute the speed of the bullet with respect to the ground, obtaining $\frac56c$, which is greater than the speed of the outlaws ($\frac34c$). Yes, the bullet reaches its target according to Galileo.

2. Denote the rapidities of the police, outlaws, and bullet with respect to the ground by $\alpha$ and $\beta$, respectively, and let the rapidity of the bullet with respect to the police be $\gamma$. Then we have \begin{eqnarray} \tanh\alpha &=& \frac12 \\ \tanh\beta &=& \frac34 \\ \tanh\gamma &=& \frac13 \end{eqnarray} In special relativity, we add the rapidities (hyperbolic angles), so that the speed of the bullet with respect to the ground is given by \begin{equation} \frac{v_b}{c} = \tanh(\alpha+\gamma) = \frac{\tanh\alpha+\tanh\gamma}{1+\tanh\alpha\tanh\gamma} = \frac57 \end{equation} so that $v_b$ is less than the speed of the outlaws ($\frac34c$). No, the bullet does not reach its target according to Einstein.

3. Performing a similar computation in each case, we obtain Table 7.5, where all speeds are given as fractions of $c$.

Table 7.5: The relative speeds of the ground, police, outlaws, and bullet with respect to each other.