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.
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To determine whether the monkey will be hit by the dart, we need to consider the path of the dart and the behavior of the monkey.
When the dart leaves the gun, it follows a parabolic path due to the effect of gravity. This means that it will initially rise and then fall back to the ground. Since the ranger aims directly at the monkey, the dart will be initially below the monkey's position as it travels towards the branch.
Now, let's consider the monkey's behavior. The monkey sees the dart leave the gun and decides to let go of the branch to avoid being hit. Since the dart is below the monkey's position, letting go of the branch causes the monkey to start falling down. If the monkey falls faster than the dart, it will be able to avoid being hit.
The velocity of the dart only affects the horizontal component of its motion, not the vertical component. Assuming the velocity is great enough to travel the distance to the tree before hitting the ground, it means that the horizontal component of the dart's velocity is sufficient.
Given this information, if the monkey falls faster than the dart, it will avoid being hit. However, if the monkey falls slower than the dart, it may still be hit. The outcome depends on the specific velocities involved.
Therefore, whether the monkey will be hit or not depends on the relative velocities of the monkey and the dart in the vertical direction.