"A person fires a bullet horizontally from a gun at the same instant that another bullet is dropped from the same height as the rifle barrel. Discuss which bullet will hit the ground first."
I'm thinking that the bullet that is dropped will hit first, because it does not have an x-component, unlike the other bullet. Am I right?
"One day in the jungle, professor Glick was stalking a rare species of monkey. He spotted one up in a tree and decided to shoot it with a dart. If the monkey begins to fall at the same instant that the dart leaves the barrel of the rifle, explain why the professor should aim directly at the monkey."
This one I do not understand. Why should he aim directly at the monkey?
Both bullets hit the ground at the same time. The vertical equation of motion is the same, and the initial downward velocity component (which determines the time to fall) is zero in both cases.
In the second question, one should aim for the monkey ONLY if the monkey is intially at the same elevation as the hunter. But that is not the situation here. The monkey is "up in a tree". You are right; the hunter should aim lower than the monkey's initial location. The "teacher", if you have one, is wrong. Congratulations.
Ah, I see! Thank you!
For the situation where a person fires a bullet horizontally from a gun at the same instant that another bullet is dropped from the same height as the rifle barrel, the bullet that is dropped will hit the ground first. Let's understand why.
Both bullets experience the force of gravity acting downward. However, the horizontally fired bullet also has an initial horizontal velocity. This horizontal velocity causes the bullet to cover some horizontal distance away from the starting point while it also falls vertically due to gravity.
On the other hand, the bullet that is dropped has no horizontal velocity. It only falls vertically due to gravity. Since both bullets experience the same downward force, the bullet that is dropped will hit the ground first because it does not have any horizontal distance to cover.
Therefore, you are correct in your thinking that the dropped bullet will hit the ground first.
Now, let's move on to the situation with Professor Glick in the jungle. If the monkey begins to fall at the same instant that the dart leaves the barrel of the rifle, the professor should aim directly at the monkey. Here's why:
When the dart leaves the barrel, it gains an initial horizontal velocity due to the force of the rifle pushing it forward. At the same time, the monkey starts to fall due to gravity. If the professor aims directly at the monkey, the horizontal velocity of the dart will ensure that it reaches the monkey as it falls.
If the professor aimed above or below the monkey, the dart would not hit it. The horizontal velocity of the dart needs to match the horizontal distance the monkey travels before hitting the ground. By aiming directly at the monkey, the dart will travel the same horizontal distance as the monkey falls, resulting in a successful hit.
Therefore, the professor should aim directly at the monkey to account for both the horizontal velocity of the dart and the vertical motion of the falling monkey.
I hope this explanation clarifies your doubts.