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The famous scenario: A hunter is trying to shoot a Monkey hanging from a tree. However, this question doesn't mention the monkey jumping down from the tree or trying to escape. (The hunter uses a tranquilizer gun, so I'm guessing the sound goes unheard.) We only know two things:

1) The height at which the monkey is hanging. 2) Distance of the monkey from the hunter.

(The hunter aims his gun directly at the monkey.)

I've solved it using this approach:

Solving for initial velocity required to launch a projectile to a given destination at a different height

Does this fit the bill or am I missing something?

Why does the answer have $d\, \text{tan}\, \theta + h$ in denominator? I'm getting $d\, \text{tan}\, \theta- h=0$ in the denominator!

Does denominator = $0$ simply mean that it's not possible? In other words, does the monkey have to fall down if the bullet has to hit the monkey?

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" ... does the monkey have to fall down if the bullet has to hit the monkey?" Yes. The previous respondent says as much. No matter how fast a projectile travels horizontally, it is always pulled to earth by gravity (see Newton's famous suggestion that a satellite could be launched into orbit around the earth on that premise). Therefore, if you aim your gun, or in archery your arrow, straight at a target you will always, quite literally, undershoot and hit the target below where you were aiming. In the monkey and hunter 'experiment' it is usually assumed that the instant the bullet leaves the gun, the monkey begins to fall. (The classic school experimental demo uses this assumption in its execution). If the bullet's horizontal velocity is sufficient it will hit the monkey because gravity is acting equally on both monkey and bullet. However, if the bullet travels too slowly it may fall short of the target and the monkey then escapes unharmed. This analysis ignores the effects of air friction on the bullet of course.

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I'll assume throughout this response that the monkey remains stationary in the tree and that there is no air resistance. If the gun is aimed directly at the monkey, then the bullet will not hit the monkey. To demonstrate this without math, I'm going to borrow part of the reasoning from user1104. If there were no gravity, and if the gun were aimed directly at the monkey, then the bullet would travel on a straight line path towards the monkey and hit the monkey. When gravity is turned on, the bullet deviates from this straight line path because of its downward acceleration, and it will end up lower than it would have with no gravity present, so it will not hit the monkey.

The only way to hit the monkey if he remains sitting in the tree is to aim above the line joining the initial position of the bullet and the monkey. If you aim high, then you can compensate for the fact that the bullet will fall during its trajectory, and this will allow you to hit the monkey.

Hope that helps!


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