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 use basic kinematics principles.

First, let's consider the vertical motion of the fragment. When the fragment is at its highest point, the vertical component of its velocity is zero. We can use the following equation to find its final vertical velocity component:

v_vertical^2 = v_initial^2 + 2gh

Here, v_initial is the initial vertical velocity component, which is zero since the fragment starts from rest at its highest point. g is the acceleration due to gravity, and h is the height at which the rocket exploded.

Therefore, we have:

v_vertical^2 = 0 + 2gh

Next, we can consider the horizontal motion of the fragment. Since no horizontal forces are acting on the fragment, its horizontal velocity component remains constant throughout its motion. Thus, the final horizontal velocity component will be the same as its initial horizontal velocity component, which we'll denote as v_horizontal.

Finally, we can determine the magnitude of the final velocity vector of the fragment using the Pythagorean theorem:

v_final^2 = v_horizontal^2 + v_vertical^2

Plugging in the values for v_horizontal and v_vertical, we get:

v_final^2 = v_horizontal^2 + 2gh

To find the angle that the final velocity makes with the horizontal, we can consider the tangent of the angle, which is equal to the ratio of the vertical velocity component with the horizontal velocity component:

tan(theta) = v_vertical / v_horizontal

Rearranging this equation, we can solve for theta:

theta = arctan(v_vertical / v_horizontal)

Substituting the values for v_vertical and v_horizontal, we get:

theta = arctan(√(2gh) / v)

Therefore, the smallest angle that the final velocity of an impacting fragment makes with the horizontal is arctan(√(2gh) / v).