My question involves broadly why calculating different (Einstein) relative velocities give different answers:
Say we have car 1 traveling in front of car 2. Car 1 goes at velocity $3c/4$, while car 2 goes at velocity $c/2$, both relative to the ground. Now, car 2 launches a projectile at car 1, with velocity $c/3$ relative to car 2. The question is, whether this projectile will reach car 1. This is a relative velocity with special relativity problem, though I am very confused why different approaches give different conclusions:
I. Compute $v_{p1}$, or the relative velocity of the projectile relative to car 1. Using the formula for relative velocities with Einstein correction, we can write $$v_{p1}=\frac{v_{p2}+v_{21}}{1+v_{p2}\cdot v_{21}/c^2}.$$ Plug in the numbers (classical relative velocities), we have $v_{p2}=c/3$, $v_{21}=-c/4$, so numerator is positive, (denominator clearly also positive), so $v_{p1}>0$, in other words the projectile hits car 1.
II. Compute $v_{pg}$, or the velocity of the projectile relative to the ground. Following the above formalism, $$v_{pg}=\frac{v_{p1}+v_{1g}}{1+v_{p1}\cdot v_{1g}/c^2}.$$ Here, we have $v_{p1}=c/3$ and $v_{1g}=c/2$ as givens in the problem. Evaluating the whole fraction, we have $v_{pg}=5c/7$, which is less than the velocity of car 1 relative to the ground $3c/4$, so projectile never hits car 1.
I just can't see what's wrong with either approach, though clearly the results are contradictory... Some guidance would be much appreciated!