A hunter at a distance dx from a coconut tree points the nuzzle of his gun directly at a monkey sitting on the tree at a height dy. The monkey sees the hunters and drops vertically from the tree at the very instant of firing. Will it be hit by the bullet? Explain.
depends. Are we considering the monkey a point mass? How far away are the hunters, and how high is the tree? "dx" and "dy" don't help much.
To determine if the monkey will be hit by the bullet, we need to consider the horizontal and vertical components of motion separately.
Let's break down the problem step by step:
Step 1: Horizontal Motion
The bullet fired by the hunter will travel in a straight line horizontally because there are no forces acting horizontally (assuming no air resistance).
Therefore, the horizontal distance, dx, has no effect on whether the monkey will be hit by the bullet.
Step 2: Vertical Motion
The monkey drops vertically from the tree, meaning its initial vertical velocity is 0 m/s, and it falls due to the force of gravity.
The height of the coconut tree, dy, is an essential factor in determining whether the monkey will be hit.
Step 3: Time of Flight
The time it takes for the bullet to reach the height of the monkey, as a result of its vertical motion, will be the determining factor.
We can calculate the time of flight using the formula for time: t = sqrt(2h / g), where h is the height and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 4: Comparing Time of Flight
If the time it takes for the bullet to reach the height of the monkey is less than or equal to the time it takes for the monkey to drop, then the monkey will be hit.
If the time of flight is greater than the time it takes for the monkey to drop, then the monkey will have already fallen by the time the bullet reaches its height, and it will not be hit.
Considering the above steps, we can conclude that the monkey will not be hit by the bullet as long as it drops from the tree before the bullet reaches its height.
To determine whether the monkey will be hit by the bullet, we need to analyze the horizontal and vertical components of the motion separately.
Let's first consider the horizontal motion. Since the monkey is dropping vertically from the tree, it does not have any horizontal velocity component. Therefore, it will continue to be at the same horizontal position during its fall.
Now let's look at the vertical motion. The monkey experiences only the force due to gravity, which causes it to accelerate downward. The time it takes for the monkey to fall from the tree is determined by the equation:
dy = (1/2) * g * t^2
where dy is the vertical distance from the tree, g is the acceleration due to gravity, and t is the time of fall.
The horizontal distance covered by the bullet during this time can be calculated using the equation:
dx = v * t
where dx is the horizontal distance from the tree, and v is the velocity of the bullet.
Since the monkey does not have any horizontal velocity, the horizontal distances covered by both the monkey and the bullet should be the same:
dx = v * t
Now, we need to compare the vertical distance (dy) of the monkey to the distance covered by the bullet (dx) during the time of fall:
dy = (1/2) * g * t^2
dx = v * t
If dx > dy, it means the horizontal distance covered by the bullet is greater than the vertical distance fallen by the monkey. In this case, the bullet will reach the monkey before it hits the ground, and the monkey will be hit.
If dx < dy, it means the horizontal distance covered by the bullet is less than the vertical distance fallen by the monkey. In this case, the bullet will not reach the monkey before it hits the ground, and the monkey will not be hit.
If dx = dy, it means the horizontal distance covered by the bullet is equal to the vertical distance fallen by the monkey. In this case, the bullet will reach the monkey at the same time it hits the ground, and the monkey will be hit.
So, to determine whether the monkey will be hit by the bullet, you would need to calculate the values of dx and dy and compare them using the above conditions.
It is important to note that this explanation assumes idealized conditions, including no air resistance. In reality, factors such as air resistance, the initial velocity of the bullet, and the accuracy of the hunter would also influence the outcome.