A hunter spies a monkey in a tree, takes aim, and fires. At the moment the bullet leaves the gun the monkey lets go of the tree branch and drops straight down. How should the hunter aim to hit the monkey?

1. Aim directly at the monkey
2. Aim high (over the monkey's head)
3. Aim low (below the monkey)

One way to do the analysis is to stick with a horizontal-vertical coordinate system and define the following:

The monkey starts a vertical distance h above the hunter, and the horizontal distance between the monkey and hunter is d. The origin is the hunter.

The horizontal position of the bullet is:

x_b = v_oxt

When x_b = d:

t

=

d vox

The vertical position of the bullet is:

y_b = v_oyt - ½ gt²

Plugging in

t

=

d vox

for the first t gives:

yb

=

d

voy vox

-

½ gt2

Compare this to the vertical position of the monkey:

y_m = h - ½ gt²

If the hunter aims the gun at the monkey to begin with:

tan(q)

=

h d

=

voy vox

So,

d voy vox

=

h

In this case, when the bullet has traveled a distance d horizontally, its vertical position will be:

y_b = h - ½ gt²

That exactly matches the monkey's position, so the bullet will hit the monkey when the hunter aims the rifle at the monkey.

That analysis was very quantitative. A more conceptual analysis is shown in the simulation above, where a tilted coordinate system is used. From that perspective it's very similar to the question we asked last time, where the ball dropped from a certain height and the ball given an initial horizontal velocity from the same height hit the ground at the same time.