Velocity addition rule in special relativity I just worked out a problem showing peculiarities of the velocity addition rule in special relativity. This was the statement for the problem:

"Nefarious thieves flee the scene of their bank robbery in a supercharged car traveling at $3/4 c$.  Hot on their tail is a police car, which pursues them at $3/5 c$.  Realizing that they will never catch up, one of the police officers fires his gun out the window at the thieves’ tires.  The bullet’s velocity (relative to the gun) is $1/5c$."

Working out the problem you get the result that, from the point of view of a stationary observer, the bullet will never reach the thieves' vehicle. My question is:
How is this resolved in the police officer's frame of reference? Will he simply see the bullet moving at a slower speed than $1/5c$ relative to himself?
 A: Another way is to use Lorentz Boost parameters (aka rapidity) which are additive.  An observer sitting still on the earth sees:
$$
\lambda_{Robbers}=tanh^{-1}({3 \over 4})=0.972955
$$
$$
\lambda_{Police}=tanh^{-1}({3 \over 5})=0.693147
$$
$$
\lambda_{Bullet}=tanh^{-1}({3 \over 5})+tanh^{-1}({1 \over 5})=0.895880
$$
Notice that  $\lambda_{Bullet} < \lambda_{Robbers}$ so the bullet never catches up to the robbers. An observer riding in the robbers' car sees:
$$
\lambda_{BulletSeenByRobbers}=\lambda_{Bullet}-\lambda_{Robbers}=-0.077075
$$
$$
v_{BulletSeenByRobbers}=tanh(\lambda_{BulletSeenByRobbers})=-0.076923c
$$
which is the velocity the robbers see the bullet moving away at.
A: So by the rule of velocity addition,
Velocity of thieves relative to lab frame = $\frac{3}{4} c = u_1$
Velocity of lab frame relative to police = $-\frac{3}{5} c = u_2$
Thus velocity of thieves relative to police = $\frac{u_1+u_2}{1+\frac{u_1u_2}{c^2}} = \frac{3}{11} c$
Since velocity of bullet relative to police = $\frac{1}{5} c $ and $\frac{3}{11} c > \frac{1}{5} c$
The thieves are faster than the bullet and the bullet can never catch them in police's frame(or in any other frame if you do similar calculations.)
Relativity works :)
