How far ....????

Question

A skydiver without a parachute jumps out of a plane 4000 ft. above the ground and quickly reaches a terminal velocity of 250 ft./sec.The skydiver's pet falcon flies back and forth between the skydiver and the ground at 500 ft/sec.How far has the falcon flown when the skydiver reaches Earth?

Answer 1

This is indeed a strange world...skydivers jumping out of planes sans parachute...falcons racing inexplicably between the skydiver and the ground. If that falcon were any kind of friend at all, he'd fly up to the plane and get a parachute. Ah well, on to your question.

The falcon is traveling twice as fast as the skydiver, so he'll reach the ground the first time in half the time it would take the skydiver. The skydiver, in the mean time, would have traveled halfway down. On the return trip, the falcon and skydiver are closing at three times the skydiver's downward speed, so they'll meet when the skydiver has fallen 1/3 of the remaining distance (1/6 of the total distance). So the falcon makes one trip down and back in the time it takes the skydiver to fall (1/2 + 1/6 =) 4/6, or 2/3, of the way down. The skydiver will have traveled 2666.67 feet so far, and the falcon will have traveled (4000 ft + 1333.33 ft =) 5333.33 feet. That's twice as far as the skydiver has fallen. (2666.67 ft x 2 = 5333.33 ft)

This scene will play out again and again until the skydiver reaches the ground. In each case, the bird will return to him when he is at 1/3 of his previous altitude, and in each case the bird will have traveled twice as far. Therefore, when the skydiver hits the ground, his psychotic, disloyal pet falcon will have traveled 8000 feet in all.

Answer 2

The simple answer is that the falcon, traveling at twice the velocity of the skydiver, travels twice the distance (8000 feet). The phrase "quickly reaches" is a bit imprecise, so the conclusion is approximate at best.

Answer 3

Don't try to solve the infinite series. The falcon is flying twice as fast, so will fly twice as far in the same period of time, assuming it takes no time at each turn around.

Answer 4

Less than 8000 ft.