The Determination of the Ideal Velocity of the Bouncing Ball

A hard rubber ball is dropped from rest. It falls to the concrete floor and bounces back up ALMOST to its initial height. A motion detector is mounted on the ceiling directly above the ball, facing down. So, the positive direction — the away-from-the-detector direction — is downward. Draw the position, velocity, and acceleration graphs.

Since moving downward is a positive direction, is the velocity positive when the ball falls, 0 when it hits the ground, and then negative when it moves up? Is the accerlation positive (9.8) as the ball falls, very positive when it hits the ground, and then neg (almost -9.8) when it moves up? Keep in mind that the problem says: the ball bounces back up ALMOST to its initial height. Also, please help me with understanding how the position graph should look like. If there is any way that you could draw me the graphs, that would really be helpful. Thank you! You’re halfway right.

Velocity will be positive as the ball falls, zero on impact, negative on the way up, and zero again at apogee.

Acceleration, however, is constantly 9.8 m/s²…the force of gravity does not change as the ball moves.

For your graphs, acceleration will be a horizontal line at y=9.8

Velocity will be a broken sinusiod (oscillating) that starts at the origin and rises to a maximum. On each impact with the ground, the graph will drop straight down to a minimum, then rise gradually through zero back up to its next maximum which will be less than the previous maximum. This will repeat until the ball is at rest.

Position will be a smooth sinusiod that begins at a maximum, falls to zero as the ball hits the ground, then rises up again to the next maximum which will be less than the previous one. This wave repeats until the ball comes to rest. This graph will never be negative, since the ball never falls below ground level. Almost everybody, at some point in their lives, has bounced a rubber ball against the wall or floor and observed its motion. Normally we don’t think about the details of bouncing ball physics too much as it’s fairly obvious what is happening — the ball basically rebounds off a surface at a speed proportional to how fast it is thrown. But what isn’t known to most is what is specifically happening to the ball before, during, and after its brief impact with the surface.

To begin this explanation let’s first consider what happens to a typical rubber ball that is dropped vertically onto a flat horizontal surface, and which falls under the influence of gravity.

In this explanation, the bouncing ball physics will be broken down into seven distinct stages, in which the ball motion (before, during, and after impact) is analyzed. To simplify the discussion 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.

Stage 1

In this stage, the ball falls vertically downward under the influence of gravity (g). The velocity V points downward. The acceleration a also points downward. The magnitude of a is equal to g, in the absence of air resistance. (Note that the acceleration due to gravity is g = 9.8 m/s2, on earth). As shown in the above figure, the horizontal force F is the friction force acting on the ball due to contact with the surface, during impact. This force F is the cause of the velocity and spin (rotation) reversal. This friction force is generated by the gripping action of the ball with the surface. The direction of this friction force is opposite the direction of slip velocity between ball and surface, during impact. Slip velocity is the relative horizontal speed between the ball’s point of impact and the surface it is impacting.

Since this force F is acting to the right, it torques the ball counterclockwise. This causes the ball to change its original spin direction from clockwise wi to counterclockwise wF, after impact. This force also causes the ball to accelerate to the right during impact. This results in the horizontal velocity component of the ball (parallel to the surface) to change direction and point towards the right, after impact.

In order for the ball velocity and spin to reverse direction it is necessary to have a high coefficient of friction between ball and surface. This creates sufficient friction force F to be generated, which causes the spin and horizontal velocity component of the ball to reverse direction after impact with the surface.