Extra Credit: There is an interesting monkey down at the zoo. The monkey spends most of its day hanging from a limb of a tree. The zookeeper feeds the monkey by shooting bananas from a banana cannon to the monkey in the tree. This particular monkey has a habit of dropping from the tree the moment that the banana leaves the muzzle of the cannon. The zookeeper is faced with the dilemma of where to aim the banana cannon in order to hit the monkey. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon? Carefully explain your reasoning and any assumptions you are making. Here are some numbers: The monkey is initially at a height of 15m, the banana cannon is 0m from the tree and shoots the bananas with an initial speed of 10m/s.
The Correct Answer and Explanation is:
Extra Credit
The zookeeper should aim the banana cannon directly at the monkey’s initial position. Since the cannon is located 0m from the tree, this means she should aim straight up toward the monkey.
The reasoning for this counterintuitive answer lies in the constant and equal effect of gravity on both the banana and the monkey. The key is to understand that from the moment the banana leaves the cannon, both objects are subject to the same downward acceleration due to gravity.
Let us first consider a hypothetical scenario without gravity. In this case, the monkey would remain motionless at its 15m height. To hit it, the zookeeper would simply aim directly at it, and the banana would travel in a perfectly straight line, eventually reaching the monkey.
Now, let us reintroduce gravity. Gravity acts on the banana and the monkey simultaneously and identically. While the banana travels upward and forward along its initial aimed path, gravity is constantly pulling it downward. At the same time, gravity is pulling the monkey downward from its branch. Over any given interval of time, gravity will cause both the banana and the monkey to fall the exact same vertical distance from their gravity free paths.
Because both the banana’s trajectory and the monkey’s position are shifted downward by the exact same amount at every moment in time, the banana will still intercept the monkey. The banana’s path curves downward from its initial aim line, but the monkey has also fallen by that same amount. In this specific case, with a vertical launch speed of 10 m/s, the collision will occur after 1.5 seconds at a height of approximately 4 meters.
The primary assumptions made are that air resistance is negligible for both the banana and the monkey, the acceleration due to gravity is constant, and the monkey lets go at the precise instant the banana is fired.
