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.
To find the smallest angle that the final velocity of an impacting fragment makes with the horizontal, we can use the principle of conservation of energy and conservation of angular momentum.
1. Conservation of Energy:
At the initial height h, the only form of energy the fragment possesses is its potential energy. As the fragment falls, its potential energy is converted into kinetic energy.
The potential energy at height h is given by: PE = mgh, where m is the mass of the fragment and g is the acceleration due to gravity.
At the point of impact, all of the potential energy is converted into kinetic energy. The kinetic energy of the fragment is given by: KE = (1/2)mv^2, where v is the speed of the fragment.
2. Conservation of Angular Momentum:
As the fragment falls, there is no external torque acting on it. Therefore, the angular momentum of the fragment is conserved.
The angular momentum L of the fragment is given by: L = mvr, where r is the distance of the fragment from the axis of rotation (which is negligible in this case).
Using the conservation of angular momentum, we can equate the initial and final angular momentum of the fragment: mvr = 0, since the final angular momentum is zero.
Now, we can find the smallest angle that the final velocity of the fragment makes with the horizontal using trigonometry.
Let θ be the angle between the final velocity and the horizontal direction.
Since the initial angular momentum is zero, the initial velocity of the fragment has both horizontal and vertical components.
The vertical component of the initial velocity is v * sin(θ).
At the point of impact, the vertical component of the velocity is zero because the fragment does not have any vertical motion.
Therefore, we have: v * sin(θ) = 0
Since sin(θ) = 0, the smallest angle θ that satisfies this equation is θ = 0.
Hence, the smallest angle that the final velocity of an impacting fragment makes with the horizontal is 0 degrees.