EXCELLENT QUESTION! I always wondered why he didn't at least TRY to shoot 'em while falling.
Well Jon does a pretty great job answering the question (although he does take a few easy shortcuts, like assuming the villain, AKA Hans, is spherical even though he's totally not):
Assume a spherical villain falling straight down.
Gravity accelerates an object at a rate of 32.174 ft/s^2 , so the downward velocity of the villain after t seconds in ft/s is 32.174 * t.
Distance traveled by a falling object is (1/2) * 32.174 * t^2 .
Hence, after 5 seconds, the villain is traveling at about 160.87 ft/s, which is not quite terminal velocity (around 180.45 ft/s), and has traveled 402.175 ft.
Now assume the villain fires straight up and is using .45 ACP FMJ ammunition. According to Wikipedia, that ammunition has an initial velocity of 830 ft/s with a test barrel length of 5 inches, and a mass of 15 g.
Since the villain and gun are traveling at 160 ft/s, the actual initial velocity of the bullet would be 670 ft/s.
Assume the bullet incurs no wind resistance. The same free-fall equations apply to decelerating the bullet.
Just eyeballing it, it would only take a bit over 0.6 seconds for the bullet to reach the villains starting point, and the bullet would only have decelerated a bit over 19 ft/s in that time. The bullet would still be traveling at about 650 ft/s, and at 15g would possess about 294 J of kinetic energy, compared to 477 J possessed by a round fired from a standstill.
After 5 seconds, then, the kinetic energy of the bullet would be decreased by about 40%. I'm guessing that would still be enough to injure someone, but I'm not sure.
What if the villain fell even further? He would reach terminal velocity of about 180.45 ft/s after about 5.61 seconds and a distance of 506.03 feet, and would continue at that velocity. A bullet fired at terminal velocity would have an initial upward velocity of about 649.55 f/s.
An object initially traveling at 649.55 f/s would take about 20.19 seconds to completely decelerate, and would have traveled about 6,557.64 feet.
The villian would fall 506.03 feet in 5.61 seconds before reaching terminal velocity, and could cover the additional 6051.61 feet in about 33.54 seconds at terminal velocity.
That means if the villain fell for just over a minute before firing back up, Bruce Willis could reach out of the window and pluck the motionless bullet from the air.
Dude. That absolutely deserves to be this week's Answer of the Week. Both Jon and Rev. Smith get a $25 gift card for Pagliacci Pizza. Congrats, sirs! I suggest you enjoy your prize while watching Die Hard. Because it is one of the best cinematic achievements of all time.
(If you have a different theory on whether or not Hans could've hit our hero while falling several stories to his death, head over to Questionland and share your smarts with the world!)