Question

. Let us examine a ball in the process of bouncing off of a floor. As we discussed briefly in class, if the ball has spin then this is a very complicated problem. So let us assume that the ball has no spin for the duration of the bounce. The ball hits the floor going vertically down.

(a) Draw a motion diagram showing the motion of the center of mass of the ball during the bounce. We are only interested in the time from when it first makes contact with the floor until the time when it leaves contact with the floor. Draw 4 points during the compression of the ball and 4 points during the re-expansion of the ball (rebound).

(b) If you haven’t done so already (all motion diagrams should have acceleration vectors...) add acceleration vectors to your motion diagram from part a). (c) Draw a force identification diagram for the ball while it is in contact with the floor, then draw a free body diagram. Remember that the ball has no spin. Don’t forget to include the direction of the net force beside your free body diagram.

(d) During the bounce is the force that the floor exerts on the ball larger, smaller or equal to the weight of the ball?

(e) Suppose the ball is going 6.00 m/s down at the instant when it hits the floor. At the instant when it leaves the floor it is going 5.00 m/s up. The whole collision with the floor takes a time of 75.0 ms. The ball has a mass of 150 g. What is the average acceleration of the ball?

(f) Use the average acceleration that you just found to get the average force that the floor exerts on the ball during the collision. (g) Bonus: Suppose everything is the same as in the previous part of the problem, except that the ball is spinning clockwise from our perspective. Suppose that because the ball is spinning the ball’s surface is slipping against the surface of the floor during the whole time it is in contact with the floor. Furthermore, assume that the coefficient of kinetic friction between the ball and floor is µk = 0.80. Find the direction of the ball’s velocity as it leaves the floor. This is a hard problem, and involves some Unit #6 ideas, but it is entirely doable using methods you learn in Units #5 and #6.

Answer 1

(a) & (b) In this explanation, the bouncing ball motion will be broken down into eight distinct stages, in which the ball motion is analyzed. Let's assume that the bounce surface is hard (rigid), and that air resistance is negligible. Let's define the geometric center of the ball as point C, the velocity of point C as V, and the acceleration of point C as a. Let's further assume that the ball has uniform density, which means that point C of the ball coincides with its center of mass. Now, I will draw 4 points during the compression of the ball and 4 points during the re-expansion of the ball (rebound).

(c) a force identification diagram for the ball while it is in contact with the floor

(d) During the bounce, the acceleration of the ball a is equal to the acceleration due to gravity and hence the force exerted by the ball on the ground is equal to the weight of the ball. The floor also exerts a reaction force according to the Newton´s third laws of force. The floor exerts a reaction force on the ball which is equal to the weight of the ball.

(e) The ball is going 6.00 m/s down at the instant when it hits the floor. At the instant when it leaves the floor it is going 5.00 m/s up. These two velocities are opposite in direction. Hence, the change of velocity will be

The acceleration will be

(f) the average force that the floor exerts on the ball during the collision

(g) Suppose everything is the same as in the previous part of the problem, except that the ball is spinning clockwise from our perspective. The kinetic frictional force (F_r) can be calculated as

Hence the frictional force gives a torque which bounce the ball to left side of our perspective