Question

A ball of mass M is thrown vertically upward with an initial speed of v_o. It experiences a force of air resistance given by F = – kv, where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M, k, v_o, and fundamental constants.

a. Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves upward? 

- increases  - decreases - remains the same Justify your answer.

b. Determine the terminal speed of the ball as it moves downward.

c. Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the height from which it was thrown?

- longer to rise - longer to fall Justify your answer. 

d. On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball's flight, where t_f is the time at which the ball returns to the height from which it was thrown.

Answer 1

a. The magnitude of the acceleration decreases as the ball moves upward. Since the velocity is upward, air resistance is downward, in the same direction as gravity. The velocity will decrease, causing the force of air resistance to decrease. Therefore, the net force and thus the total acceleration both decrease. 

b. At terminal speed ΣF = 0. ΣF = –Mg + kv_T giving v_T = Mg/K

c. It takes longer for the ball to fall. Friction is acting on the ball on the way up and on the way down, where it begins from rest. This means the average speed is greater on the way up than on the way down. Since the distance traveled is the same, the time must be longer on the way down.