A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal. (Answer using g, h, and v, as needed.)

Question

A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal. (Answer using g, h, and v, as needed.)

Answer 1

To find the smallest angle that the final velocity of an impacting fragment makes with the horizontal, we can break down the problem into two parts:

1. Find the time it takes for the fragment to fall from the peak height h to the ground.
2. Use this time to calculate the horizontal and vertical components of the final velocity of the fragment.

Let's go through each part step by step:

1. Find the time it takes for the fragment to fall from the peak height h to the ground.
- The vertical motion of the fragment is governed by free fall under the influence of gravity (assuming no air resistance). The equation for the height h as a function of time t is given by:
h = (1/2) * g * t^2
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Rearranging the equation, we get:
t = sqrt(2h / g)
- This gives us the time it takes for the fragment to fall from the peak height h to the ground.

2. Use the time to calculate the horizontal and vertical components of the final velocity of the fragment.
- The horizontal component of the final velocity will remain constant, as there is no acceleration in the horizontal direction. Therefore, the horizontal component of the final velocity is equal to the initial horizontal velocity of the fragment, which is v.
- The vertical component of the final velocity can be calculated using the equation:
v_vertical = g * t
Where t is the time calculated in step 1.
- Now that we have the horizontal and vertical components of the final velocity, we can use trigonometry to find the angle. The angle θ can be calculated using the equation:
tan(θ) = v_vertical / v_horizontal
Where v_vertical is the vertical component of the final velocity and v_horizontal is the horizontal component of the final velocity.

By following these steps, you can calculate the smallest angle that the final velocity of an impacting fragment makes with the horizontal using the given values of g, h, and v.