A park ranger wants to shoot a monkey hanging from a branch of a tree with a tranquilizing dart. The ranger aims directly at the monkey, not realizing that the dart will follow a parabolic path and thus will fall below the monkey. The monkey, however, sees the dart leave the gun and lets go of the branch to avoid being hit. Will the monkey be hit anyway? Does the velocity of the dart affect your answer, assuming that it is great enough to travel the horizontal distance to the tree before hitting the ground? Defend your answer.
Yes (it will be hit anyway) and no (the dart velocity does not matter. The dart fall below the direct-aim line by the same distance that the monkey has fallen, at any time before the dart hits the ground.
The monkey will not be hit by the dart. The reason is that the monkey sees the dart leave the gun and lets go of the branch, which means it starts to fall vertically downwards. Meanwhile, the dart follows a parabolic path due to its horizontal velocity and gravitational acceleration.
Since the monkey is falling vertically downwards, the dart will take some time to reach the height at which the monkey was initially hanging. By this time, the monkey will have fallen below that height.
The velocity of the dart does not affect this answer. Regardless of its velocity, the dart will still follow the same parabolic path. However, if the dart's velocity is great enough to travel the horizontal distance to the tree before hitting the ground, it is possible that the dart could hit the tree itself, but not the monkey.
To determine whether the monkey will be hit by the tranquilizing dart, we need to consider the key factors at play: the initial velocity of the dart, the angle at which the dart is shot, the height of the tree branch, and the time it takes for the dart to reach the branch.
When the ranger shoots the dart, it will follow a parabolic path under the influence of gravity. The projectile motion of the dart can be broken down into independent horizontal and vertical motions.
Since the dart is aimed directly at the monkey, the horizontal velocity of the dart will be constant throughout its motion and will not affect the outcome. As long as the velocity is great enough for the dart to reach the horizontal distance to the tree before hitting the ground, we can assume that it will reach the tree.
On the other hand, the vertical motion is affected by gravity. The dart will experience linear acceleration downward due to gravity, which will cause it to fall below the monkey's initial position.
Now, considering that the monkey sees the dart leave the gun and lets go of the branch, the monkey will start to fall the moment it releases the branch. Given that the projectile motion equations have symmetry, the time it takes for the dart to reach the height of the branch is the same as the time it takes for the monkey to fall to the same height.
Therefore, as long as the dart reaches the tree branch before the monkey falls to that height, the monkey will not be hit by the dart. This outcome remains independent of the velocity of the dart, assuming it is sufficient to reach the tree.
In summary, the monkey, by letting go of the branch, will avoid being hit by the tranquilizing dart, regardless of the dart's velocity, as long as the dart can reach the horizontal distance to the tree before hitting the ground.