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?
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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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