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Register a new userMotions of a billiard ball after a cue stroke
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Authors
Hyeong-Chan Kim
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We study the collision between the cue and the ball in the game of billiards. After studying the collision process in detail, we write the (rotational) velocities of the ball and the cue after the collision. We also find the squirt angle of the ball for an oblique collision which represents the deviation of the ball from the intended direction.
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We study some statistical properties for the behavior of the average squared velocity -- hence the temperature -- for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic leading the average squared velocity to reach a steady state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and, (ii) billiard dynamics using either numerical simulations and theoretical description.
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Some dynamical properties of a bouncing ball model under the presence of an external force modeled by two nonlinear terms are studied. The description of the model is made by use of a two dimensional nonlinear measure preserving map on the variables velocity of the particle and time. We show that raising the straight of a control parameter which controls one of the nonlinearities, the positive Lyapunov exponent decreases in the average and suffers abrupt changes. We also show that for a specific range of control parameters, the model exhibits the phenomenon of Fermi acceleration. The explanation of both behaviours is given in terms of the shape of the external force and due to a discontinuity of the moving walls velocity.
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