# Friction on a rolling ball

How long will a ball, say a billiard ball, stop rolling on a flat surface for a given coefficient of friction? What is the formula for the ball's decreasing velocity?

## 1 Answer

 Time to stop is  initial_speed/(mu * g)


Derivation below.

Assume the ball is moving to the right, then the free body diagram is The equation of motion is $$m\ddot{x} = -\mu N$$ But $N = mg$ hence $$\ddot{x} = -\mu g$$ Using the constant acceleration kinematic equation $$\dot{x}_{final} = \dot{x}_{initial} + \ddot{x}t = \dot{x}_{initial} - \mu gt$$ Then, by letting final speed be zero, we solve for the time for the ball to stop. $$t= \frac{\dot{x}_{initial}}{\mu g}$$ For example, using coefficient of friction $\mu = 0.15$, initial speed of $\dot{x} = 15$ meters per second and earth's gravity $g=9.81$, then the time to stop is $$t = \frac{15}{(0.15)(9.81)} = 10.2 s$$

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