Deformation of ball in elastic collision

[Beer-monster]

Homework Statement

A soccer ball with radius R = 11 cm is inflated to a gauge pressure of 9×104 Pa. The ball is dropped onto and bounces elastically off of a hard smooth floor. Find approximate expressions for the surface area of the ball in contact with the floor, the amount of time the ball is in contact with the floor, and the peak force exerted on the floor, if the mass of the ball is 0.42 kg and it is dropped from a height of 0.1 m onto the floor.

Homework Equations

Potential energy of falling object [tex]mgh[/tex]

Conditions of perfectly elastic collision: [tex] \frac{mv_{i}^{2}}{2} = \frac{mv_{f}^{2}}{2}[/tex]

The Attempt at a Solution

I think I can calculate the change in volume of the ball by equation the kinetic energy of the ball on contact with the floor (equal to the original potential energy mgh) with the work done to compress the ball against the internal pressure i.e PdV.

After this I'm lost. How can I calculate the fraction of a spheres surface in contact with the flat surface and relate this to the change in volume?

I think if I knew this I might be able to calculate the time the ball is in contact with the surface by considering the change in momentum when the ball bounces back and then using the equation for impulse to work out the time scale for this relationship.

Any help would be appreciated.

 

[anathema]

Hi there,

It looks like you're on the right track with your approach! To calculate the fraction of the sphere's surface in contact with the floor, you can use the fact that the area of contact is equal to the change in volume divided by the radius of the sphere. This can be derived from the definition of a sphere's volume and surface area, or you can also use the formula for the volume of a spherical cap.

Once you have the area of contact, you can use the formula for impulse to find the time the ball is in contact with the floor. Remember that the impulse is equal to the change in momentum, and you can use the formula for the change in momentum in an elastic collision to find the peak force exerted on the floor.

Hope this helps! Let me know if you have any further questions.