What Are the Physics behind Bouncing Balls?

Let's break down the physics of bouncing balls.

To begin, we'll look at the simplified seven stages of a ball bounce, ignoring any outside force other than gravity. We'll break down each step in detail below with equations, but if you need a deeper visual, the video below will break that down too.

Stage 1: Falling

Stage one is the begging of every ball bounce, where potential energy from the height of the ball is converted into kinetic energy through acceleration due to gravity. In a simplified case, the ball falls in line with the force of gravity, which always points directly downward. On Earth, this acceleration due to gravity is 9.8 m/s2 (g= 9.8 m/s2). This means, in essence, that for every second of falling, the ball's velocity will accelerate by 9.8 m/s.

Stage 2: Initial contact

The initial contact phase is just that; when the ball barely touches the ground surface. It will continue to fall under the influence of gravitational acceleration, but now, a normal force from the ground surface, opposing the force due to gravity, will act on the ball. Stage 3: Deceleration/negative acceleration.

After the initial impact, the ball rapidly decelerates or rather accelerates in a negative direction. The ball's velocity still points downward as it deforms, but acceleration on the ball begins to point upward as the forces from the reaction overcome gravity. This all means that the ball is pushing on the ground with force greater than its own weight, so acceleration must point upward.

Stage 4: Maximum deformation

Following the deceleration stage, the ball has reached maximum deformation. The velocity is zero at this point, and the acceleration vector points upward. This is the lowest point of the ball, as well as its maximum deformed point. If we assume the ball to be totally elastic and ignore other energy losses like sound and heat, then the ball would bounce back up to its original drop height after this point.

Stage 5: Initial rebound

This stage begins the ball's journey back to where it began. Its velocity and acceleration vectors point in the same direction, meaning upward movement. The ball is less deformed than the maximum deformation stage, and due to its elasticity, it is now pushing against the surface with a force greater than its own weight. This is what will cause the ball to bounce upward.

Stage 6: Zero contact rebound

The ball is no longer deformed at zero contact rebound and barely touches the surface, essentially only at one point. Velocity is moving the ball upward, but at this point, acceleration switches to oppose the velocity vector.

This is because there is no longer any force from the elasticity of the ball pushing on the surface, giving it an upward acceleration. Acceleration due to gravity pulling downward will now be the only force acting on the ball in a perfect system.

Stage 7: Full rebound

At full rebound, the ball has left the surface, and its velocity vector still points upward, though shrinking steadily due to the acceleration or deceleration due to gravity. Following this step, the ball will peak at a new step, where its velocity vector is zero, and gravity is the only force acting on it.

Added variables and special cases in bouncing ball physics

The case of the bouncing ball above was simplified to remove other forces like air resistance, imperfect elasticity, spin, friction, and the force from an initial throw, among others. All this means that bouncing ball physics gets more complicated from here.

When balls have any spin, as they usually do when thrown, and when the surface they hit isn't frictionless, the ball's spin reverses from before to after impact. This is due to the force of friction. Assuming 2-dimensions for theory's sake, you can observe the reaction below.

As the ball impacts with a spin in one direction, the friction force F counteracts the ball's spin. Or rather, the friction force is always opposite the direction of the slip velocity between the spinning ball and the surface. Since the friction force is opposite of the ball's spin, it torques the ball in the other direction. It also causes the path of the ball's bounce to skew in the direction of the friction force. In simplified terms, when a ball spins in one direction when it hits a wall, the friction between the ball and the wall overcomes the spin so much that it reverses its spin direction.

This spin reversal doesn't happen if the ball and the wall's coefficient of friction aren't high enough. The coefficient of friction varies by material and surface and is essentially a number that indicates how grippy a surface or material is.

In real-life non-ideal scenarios, bouncing balls lose energy and eventually come to a stop. This is all due to the forces we ignored in the first example. When a ball hits a wall or surface, it makes a noise, a loss of energy from the bounce. It also will generate some amount of heat, another loss of energy. Friction from the wall will cause energy loss and air resistance while the ball travels.

In essence, the ball will never have as much potential or kinetic energy as it had from right after it was thrown or right before it strikes a surface, depending on the scenario.