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A solid ball of mass
*m*, size
*r* and spin
*ω* about an axis through its center is dropped freely from a height h on a rough horizontal plane. Assuming its angular momentum is parallel to the horizontal plane upon impact it bounces repeatedly drifting on a vertical plane. We analyze the kinematics of the bouncing ball assuming the impacts are semi-elastic without slipping. By varying the spin and relevant parameters, a robust
*Mathematica* [1] program enables simulating the trajectories.

Kinematics, including multiple trajectories of a bouncing massive point-like particle on a vertical plane thrown at an angle with respect to a horizontal surface is a classic physics problem [

At the impact the floor imparts three different effects: 1) it reorients the initial downward vertical velocity in a slanted upward direction 2) because of surface roughness it generates a horizontal impulse drifting the ball along the horizontal, specifically, it pushes the ball along the opposite direction of the orientation of the angular velocity at the impact, x-axis as shown in

It is intuitive to say that right after the bounce because gravity is the only active force acting on the ball the center of the ball would trace a parabolic trajectory. Furthermore, because of the same reason the spin of the ball stays the same; its angular momentum conserves. One may also extend the aforementioned conclusions for all subsequent multiple bounces the number of which is determined by the restitution factor, e.

With this intuitive insight to quantify the kinematics we formulate the problem.

We write the dynamic version of the Newton’s law as,

where _{2x}. Accordingly, Equation (1) along the x-axis is,

one may realize the integral on the Left Hand Side (LHS) is the linear impulse along the y-axis. Note also the orientation of the static friction shown in

where _{cm} is the moment of inertia of the rotating object about the center-of-mass (cm) and ω’s are the associated spins, i.e. angular velocities. Applying Equation (3) to the case on hand replaces the integrand with

Substituting Equation (2) in (4), gives,

Furthermore, while the ball is in contact with the floor and rolls without slipping we apply,_{2x}, right after the bounce is subject to

Accordingly, we realize the explicit relationship between the spins before and after the impact; i.e., _{cm}, may be substituted as

_{cm}, Equation (6) yields,

its value for a solid ball is,

In general the ball bounces more than once. Following analysis similar to the aforementioned reveals the subsequent bounces don’t alter the spin other than the first bounce; i.e. the spin given by Equation (7) stays the same. This is justified according to Equation (5). For a multiple bounce it reads,

where i and f indicate the “initial/before” and “final/after” states. For a rolling ball we substitute

For instance, the runtime between the first and the second impact is,

During this time interval

Its value for a solid ball with

mentioned in [

Having considered the aforementioned information we derive analytic equations describing the trajectories of a spinning bouncing ball. In the coordinate system depicted in

In these equations g is the gravity, the takeoff speed after the first impact is v_{2}, the projectile (reflected) angle is θ, h is the initial height and the restitution factor is, e.

Utilizing this information we craft a Mathematica code simulating the features of a spinning bouncing elastic ball. Accordingly, as shown in

Auxiliary information about the quantities of interest such as range, Equation (9), for a chosen initial spin ω vs. the number of the bounces is shown in

For instance as shown, a spinning ball with ω = 9 rad/s dropped form a 3.0 m height drifts 27.9 cm between the first and the second impact; 19.6 cm between the second and the third and etc.

Linear impulse and angular impulse are useful applied mechanical quantities, yet their applied mathematical models quantifying their applications are somewhat hindered in the literature. In a real-life situation one may encounter cases such as, dribbling a spinning basketball or throwing a spinning tennis ball on a floor. Mathematical challenges analyzing these kind of problems stem mainly from the size of the object. In elementary scenarios, the size is ignored suppressing the degrees of freedom reducing the problem to analysis of point-like objects. Reference [

Sarafian, H. and Lobe, N. (2017) Angular Impulse and Spinning Bouncing Ball. World Journal of Mechanics, 7, 177-183. https://doi.org/10.4236/wjm.2017.77016