Monkey and Hunter

A hunter spies a monkey in a tree and wants to shoot it. The hunter knows that at the instant he fires the gun the monkey will let go of the tree branch and drop straight down. How should the hunter aim to hit the monkey? The monkey starts a vertical distance h above the hunter and the horizontal distance between the monkey and hunter is d (see transparency).

Aim directly at the monkey
Aim high (over the monkey's head)
Aim low (below the monkey)

Solution: Exploit independence of x and y motion and apply the kinematic equations!

Bullet motion:

Horizontal position: x_B(t) = v_x(0) t
Impact time (defined by x_B = d): t_Impact = d/v_x(0)

Vertical position: y_B(t) = v_y(0)t -½gt²
Vertical position at impact: y_B(t_Impact) = v_y(0)t_Impact -½gt_Impact²

Monkey motion:

Vertical position: y_M = h - ½gt²

Aim at monkey means: tan θ=v_y(0)/v_x(0)=h/d h= v_y(0) d/v_x(0)

Vertical position of monkey is: y_M = h - ½gt² = v_y(0) d/v_x(0) - ½gt²
Vertical position at impact: y_M(t_Impact) = v_y(0) d/v_x(0) - ½gt_Impact² = v_y(0) t_Impact - ½gt_Impact² = y_B(t_Impact)

Monkey is hit!