A park ranger shoots a monkey hanging from a branch of a tree with a tranquilizer 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 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

To determine whether the monkey will be hit by the dart or not, we need to consider the motion of both the monkey and the dart.

When the ranger shoots the dart, it will follow a parabolic path due to the force of gravity acting on it. The dart will initially start moving horizontally but will gradually start to fall towards the ground as it travels forward. This curved path is known as a projectile motion.

Now, let's analyze the monkey's motion. As soon as the monkey sees the dart gun, it decides to let go of the branch and fall downwards. Since the monkey falls freely under the influence of gravity, its trajectory will also be a parabolic path.

For the monkey to be hit by the dart, the paths of the monkey and the dart must intersect at some point in space. It is important to note that the motions of the monkey and the dart are independent of each other. The monkey's motion is solely determined by gravity, while the dart's motion is determined both by gravity and the horizontal velocity imparted to it by the ranger.

If the monkey falls down at the exact same rate as the dart, accounting for the horizontal velocity of the dart, the paths will intersect at the same time, and the monkey will be hit. However, if the monkey's vertical speed is greater than the vertical speed of the falling dart, it will fall faster and move out of the way of the dart.

The horizontal velocity of the dart does not affect whether the monkey will be hit or not. It only affects the horizontal distance the dart will travel before hitting the ground. As long as the dart has enough velocity to reach the tree horizontally, it can still hit the monkey if their trajectories intersect in the vertical direction.

In summary, if the monkey falls faster than the dart's vertical speed, it will avoid being hit. The horizontal velocity of the dart does not change the outcome unless it impacts the monkey's trajectory in the vertical direction.