[PhD Qualifier] Hockey puck friction

[confuted]

This one seems like it should be easy, not sure where the trouble is.

Homework Statement

A circular ice rink lies in a horizontal plane. A puck of mass M is propelled from point A along the rail of the ice rink so that the puck moves in a circular path. The magnitude of the initial tangential velocity is $$v_0$$. The rail exerts a frictional force $$\mu F_c$$ on the puck causing the velocity $$v(t)$$ to decrease with time, $$t$$. The magnitude of the centripeal force is $$F_c$$ and $$\mu$$ is the coefficient of friction between the puck and the rail. The radius of the ice rink is $$R$$. Assume there is no friction between the puck and the ice.

a) Calculate the speed $$v(t)$$ of the puck.
b) Calculate the total distance the puck will travel from t=0 to $$t=\infty$$, i.e. $$s=\int_0^\infty v(t)dt$$

Homework Equations

$$a_c=\frac{v(t)^2}{R}$$

The Attempt at a Solution

$$F_f=\mu F_c=\mu M a_c = \frac{\mu M v^2}{R}$$
$$\frac{dv}{dt}=-a_f=-\frac{F_f}{M}=-\frac{\mu v^2}{R}$$
$$\frac{dv}{v^2}=-\frac{\mu dt}{R}$$
$$\int_{v0}^{v}\frac{dv}{v^2}=-\int_0^t\frac{\mu dt}{R}$$
$$\frac{1}v-\frac{1}{v_0}=\frac{\mu t}{R}$$
$$v=\frac{R v_0}{R+\mu v_0 t}$$

Now this result must be incorrect, because
$$s=\int_0^{\infty}{\frac{R v_0}{R+\mu v_0 t}dt}=\frac{R}{\mu}\ln(R+\mu v_0 t)=\infty$$ ... nonsense

Where did I go wrong?

 

[tiny-tim]

Now this result must be incorrect, because
$$s=\int_0^{\infty}{\frac{R v_0}{R+\mu v_0 t}dt}=\frac{R}{\mu}\ln(R+\mu v_0 t)=\infty$$ ... nonsense

Where did I go wrong?

Hi confuted!

Looks ok to me …

if the speed goes down by 10, the deceleration goes down by 100 …

why shouldn't it travel infinitely far?

 

[confuted]

It just seems nonphysical -- are you sure I haven't made some mistake?

 

[tiny-tim]

It just seems nonphysical -- are you sure I haven't made some mistake?

"nonphysical"? …

what about good ol' Newton's first law … isn't that physical??!

Physical things do carry on for ever unless there's some good reason not to!