A ball climbing a ramp while "rolling the wrong way"

[FranzDiCoccio]

You can get around that with calculus.
$\Delta L=I\Delta\omega=r\int F.dt$
$\Delta p=m\Delta v=\int F.dt$.
Eliminate $\int F.dt$.

Sure, I know. But 17 yr old students won't know integrals for a couple of years or so. I meant harder for them.

On second thoughts, they do have the concept of average. So for them we say: even if friction changes along the way, by definition its overall effect can be described through an average friction
$\Delta L=I\Delta\omega=r \bar F \Delta t$
$\Delta p=m\Delta v=\bar F \Delta t$.
Eliminate $\bar F \Delta t$

This is actually what I was referring to when I mentioned "average friction force and torque" a few posts back [post-6492781].

It would be very rare to worry about acceleration being discontinuous.

Well, at the end of the frictionless surface the discontinuity is perhaps to be expected. But I find the other one, when the ball starts rolling, kind of intriguing.
For some reason I expect the speed to vary smoothly between v_0 and v_1.

 

[haruspex]

Well, at the end of the frictionless surface the discontinuity is perhaps to be expected. But I find the other one, when the ball starts rolling, kind of intriguing.
For some reason I expect the speed to vary smoothly between #v_0# and #v_1#.

How about a sliding object coming to a stop? That's usually taken as a sudden transition from a constant-ish nonzero acceleration to a zero one. Of course, one could allow that no object is perfectly rigid, so it doesn’t all stop at once, etc., ultimately getting down to oscillations of interacting dipoles. But does the standard simplification keep you awake at night?

 

[FranzDiCoccio]

Ok, I see... The point is that acceleration is caused by kinetic friction, which vanishes as soon as the object stops.

But it works even if the object does not stop, e.g. because the horizontal plane it is sliding on has a "rough patch" of finite length, that is not sufficient to stop it.

Here too the speed decreases linearly, but won't vanish.

Something similar happens with the sliding ball, although it's a bit more subtle.
When the object attains the "rolling without slipping" speed, the kinetic friction vanishes even though the surface has no discontinuity.
The relevant friction is static, which does not produce work, and hence it does not slow down the object further.Cool. Thanks