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?

 Time to stop is  initial_speed/(mu * g)

Derivation below.

Assume the ball is moving to the right, then the free body diagram is

Mathematica graphics

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 $$ t = \frac{15}{(0.15)(9.81)} = 10.2 s$$

  • \$\begingroup\$ Latex is not supported. Some of the stack exchange sites support MathJax, but GDSE is not currently one of them. There's a meta post regarding adding MathJax support to GDSE. If it would have helped your question, vote accordingly. \$\endgroup\$ – Pikalek Oct 10 '17 at 2:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.