Physics: ball launch using gravity

[Tom Bottiglieri]

Originally Posted by David Brinza

Using conservation of energy, you should be able to determine from what height you will need to start rolling the ball down a "ski jump" in order to achieve enough velocity to launch the ball at some angle to make it through the hoop. Don't forget to include the rotational energy of the ball in your calculation...

As David said...

Total energy is a system is constant.

Potential Energy (U)  = (mass) * (height) * (acceleration of gravity)
Linear Kinetic Energy (T) = 1/2 * (mass) * (velocity ^ 2)
Rotational Kinetic Energy (KR) = 1/2 * (moment of inertia) * (velocity ^ 2)

I for a sphere: 2/5 * Mass * Radius^2

EA = EB (E is total energy, A and B are arbitrary points in time.)

E = T + U + KR

So what I would do first is a simple kinematics/projectile motion problem (there might be some handy tools in the white paper section ) to find what velocity is needed on the launch. (In this case, you would arbitrarily chose the angle theta of the launch. I would chose 45 degrees.)

Once you have your needed velocity and launch angle, figure out how much kinetic energy it will have at launch. Plug the velocity, mass, and Moment of Inertia (I) into the equations above, and add the 2 values together to find net kinetic energy. Then, set that equal to the potential energy (You can assume this if your reference frame has the balls launch point as (0,0) and the ball is starting from a stop, aka no kinetic energy.)

So now you have U = T + K_R and U = mgh. Simply solve for U, and divide by 0.098 (g * m) to find how high (in meters) your ball should start in the y direction above the launch point.

Voila.