A hunter aims directly at a monkey hanging from a tree branc

A hunter aims directly at a monkey hanging from a tree branch. If the monkey releases his grip the instant the hunter fires, will the monkey be killed? If Yes, then t=?

Solution

Let:
D = horizontal distance from hunter to monkey;
H = Monkey\'s initial height.
V = bullet\'s initial speed.

That means the gun is aimed at an angle given by:

tan = H/D

Next, calculate the bullet\'s y-coordinate as a function of time (use the hunter\'s position as the origin:

y_b(t) = (Vsin)t - ½gt²

Calculate the monkey\'s y-coordinate as a function of time:

y_m(t) = H - ½gt²

Determine at what moment \"T\" the bullet and monkey will be at the same height:

y_b(T) = y_m(T)
(Vsin)T½gT² = H½gT²
(Vsin)T = H
T = H/(Vsin)

Determine the bullet\'s horizontal position \"X\" at time T:

X = x_b(T) = (Vcos)T
= (Vcos)(H/(Vsin))
= H/tan()
= H/(H/D)
= D

In other words: At time \"T\", the bullet is: (1)at the same height as the monkey; and (2)at the same horizontal location as the monkey (namely \"D\"). In other words, the bullet is at the same location as the monkey. Bad news for the monkey.