finding angular velocity and regular velocity when bouncing off a surface with friction?

Question

Take the game of pong as a simple example. When you hit the ball with a paddle that has a frictional surface, the ball will spin as well as change direction according to the coefficient of kinetic friction on the paddle and the velocity of the paddle. The ball will then spin, and when it hits another surface this spin will cause a change in direction as well. Assuming that the velocity of the paddle is never low enough to use the static coefficient of friction of the paddle, how would i find the generated angular velocity on the ball and the new velocity of the ball?

Answer 1

This is a very complex problem, but let's tackle it step by step. To simplify things let's take the paddle to be motionless (A strange concept, but much easier to think about). So all of the motion will be in the ball. This motion will be at some angle θ with respect to the horizontal (here parallel to the paddle surface). The total energy of the system will remain conserved, so we will have before collision with the paddle KE = 1/2*m*v_i², and then the energy after collision will be E = 1/2*m*v_f² + 1/2*I*w² - F_friction*d. where w is the angular velocity.

So to find w we need to have the initial velocity (relative to the paddle), the mass of the ball (assumed given), the final velocity (also relative to the paddle), the force of friction (μ*Normal), the distance traveled along the paddle during contact, the moment of inertia for the ball (if it is a hollow ball then it is 2/3 m*R²), and the radius of the ball.

Whew! that is a lot of information! I presume that μ is given, that m is given, that v_i is given, and that R is given. I will additionally assume that the normal force is constant during the ball's travel along the paddle and that its mass is sufficiently small that the force of gravity can be ignored for this interaction (while not valid for the entirety of the travel, it seems reasonable during contact from experience). therefore we can find the normal force via the change linear momentum along the y-direction p_y = m*v_i*sin(θ) and the change in time of the impact Δt. so let F_Norm = (m*v_f*sin(Φ) - m*v_i*sin(θ))/ Δt. where Φ is the angle that the ball leaves with.

With this information in tow we must simply apply linear dynamics F = ma, such that F_friction = μ*F_Norm = m*(v_f - v_i)/Δt

This leaves us with the two unknowns v_f and w. without further information it is hard to proceed, but hopefully this is a good starting point!

Hope that this has helped. And, remember, never stop learning!