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 answer these questions, we can use the basic principles of momentum and impulse.
(a) The magnitude of momentum (p) before the collision can be calculated using the formula p = m * v, where m is the mass of the ball and v is the velocity. Since the ball is dropped from rest, its initial velocity (vi) is 0. Therefore, the magnitude of momentum before the collision is:
p_before = m * vi = m * 0 = 0
(b) The magnitude of momentum after the collision can also be calculated using the formula p = m * v. In this case, the ball bounces back up to a height hf. Since the ball reaches the highest point and then starts to descend, its final velocity (vf) can be calculated using the formula vf^2 = vi^2 + 2 * g * (hf - hi), where g is the acceleration due to gravity. Rearranging this formula, we get vf = sqrt(vi^2 + 2 * g * (hf - hi)). Therefore, the magnitude of momentum after the collision is:
p_after = m * vf = m * sqrt(vi^2 + 2 * g * (hf - hi))
(c) The magnitude of the impulse imparted to the ball is given by the change in momentum, which can be calculated using the formula impulse = p_after - p_before. Substituting the previously calculated values, we get:
impulse = p_after - p_before = m * sqrt(vi^2 + 2 * g * (hf - hi)) - 0 = m * sqrt(vi^2 + 2 * g * (hf - hi))
(d) The magnitude of the average force of the ground on the ball can be calculated using the formula average force = impulse / tc. From part (c), we know the value of impulse. Therefore, substituting the values, we get:
average force = impulse / tc = (m * sqrt(vi^2 + 2 * g * (hf - hi))) / tc
Note that the answer is in terms of m, hi, hf, tc, and g as requested.