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?
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:
xb = voxt
When xb = d, t = d/vox
The vertical position of the bullet is:
yb = voyt - 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:
ym = 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:
yb = 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.