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Dr Ian Plummer

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Technical
Properties of Sliding and Rolling Croquet Balls

When a ball is struck it starts moving by sliding over the grass. As it travels the friction between the ball and the grass causes it to start rolling until the rate of rolling is matched to its progress across the grass, that is, there is no sliding. What is shown below is the kinetic energy of a rolling ball is 5/7ths the kinetic energy of a sliding ball with equal energy.  Alternatively this can be expressed as the velocity of a rolling ball travels at ~84% of the velocity of a sliding ball struck with equal force.

A consequence of this is when balls collide. When a moving ball collides with (roquets) a stationary ball it is only the kinetic energy, not the rotational energy, which is partitioned between the two balls. Energy locked up in the rolling motion is not transferred to the stationary ball.

Consider a ball with an initial sliding velocity of u. The total energy, ET , of the ball is solely its kinetic energy, EK , the energy due to its skidding or sliding velocity. The kinetic energy is given by the following:

formula

The mass of the ball is given by m. For a rolling ball with linear velocity v, there are two components of energy; the kinetic energy as above and the rotational energy, ER. The rotational energy for a body is given by the following equation, and the components of it follow.

formula

I is the moment of inertia. For a solid sphere, i.e. the croquet ball, this is given by:

formula

Again m is the mass of the ball and r is its radius. 'w' is the rotational velocity in radians per second:

formula
Thus ER is:
formula

For the rolling ball its total energy is the sum of the kinetic and rotational energy:

formula

formula

We can now compare the linear velocities of two balls hit with the same energy, one sliding and the other rolling. As ET is the same,

formula

formula

At this point, given that kinetic energy is proportional to the square of the velocity, we can see that the kinetic energy of the rolling ball is 5/7 (71.4%) that of a sliding ball.

Taking square roots yields the relative velocities:

formula

Or, the rolling velocity is 84.5% of the skidding velocity.

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Updated 1.i.11
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