A superball of mass m, starting at rest, is dropped from a height hi above the ground and bounces back up to a height of hf. The collision with the ground occurs over a total time tc. You may ignore air resistance.
(a) What is the magnitude of the momentum of the ball immediately before the collision? Express your answer in terms of m, hi, and g as needed (enter h_i for hi, h_f for hf, and t_c for tc).
(b) What is the magnitude of the momentum of the ball immediately after the collision? Express your answer in terms of m, hf, and g as needed (enter h_i for hi, h_f for hf, and t_c for tc).
(c) What is the magnitude of the impulse imparted to the ball? Express your answer in terms of m, hi, hf, tc, and g as needed (enter h_i for hi, h_f for hf, and t_c for tc).
(d) What is the magnitude of the average force of the ground on the ball? Express your answer in terms of m, hi, hf, tc, and g as needed (enter h_i for hi, h_f for hf, and t_c for tc).
To solve this problem, we can use the principles of conservation of momentum and the impulse-momentum theorem.
(a) The magnitude of the momentum of the ball immediately before the collision can be calculated using the equation p = m * v, where p is the momentum, m is the mass, and v is the velocity. Since the ball is dropped from rest, its initial velocity is zero. Therefore, the magnitude of the momentum before the collision is:
p_before = m * v_before = m * 0 = 0.
(b) The magnitude of the momentum of the ball immediately after the collision can also be calculated using the equation p = m * v, where m is the mass and v is the velocity. Since the ball bounces back up to a height of hf, its final velocity can be determined using the equation v_final = sqrt(2 * g * hf), where g is the acceleration due to gravity. Therefore, the magnitude of the momentum after the collision is:
p_after = m * v_after = m * sqrt(2 * g * hf).
(c) The magnitude of the impulse imparted to the ball is equal to the change in momentum, which can be calculated using the equation impulse = p_after - p_before. Substituting the values from parts (a) and (b), we get:
impulse = p_after - p_before = (m * sqrt(2 * g * hf)) - 0 = m * sqrt(2 * g * hf).
(d) The magnitude of the average force of the ground on the ball can be calculated using the impulse-momentum theorem, which states that impulse is equal to the average force multiplied by the time of collision. Therefore, the magnitude of the average force can be calculated as:
average_force = impulse / tc = (m * sqrt(2 * g * hf)) / tc.
This formula gives the magnitude of the average force exerted by the ground on the ball during the collision.