Kinematics of ball rolling motion

[rela]

Hi everyone,

I'm kinda new and I'm trying to understand some kinematic principles which I can't really figure out...

Without using intuition, what is the proof behind in explaining that the velocity of a rolling ball down a straight ramp increases and the velocity decreases when the rolling ball goes up a straight ramp?

Intuitively, I know it's like that but I just want to know the concept and proof behind this.

Would be grateful if anyone could explain this concept in detail.

Thanks!

 

[Doc Al]

What aspect of this are you having trouble with?

You can understand the acceleration of the rolling ball by applying Newton's 2nd law (analyzing the forces acting on it) or by applying conservation of energy.

 

[rela]

Hi,

Appreciate your revert. Err...Perhaps the attachment above can outline my confusion better... What I'm seeing is a ball set to rolling on a track (horizontal->downward straight ramp->horizontal->upward straight ramp).

At t=t0 it is released at the very left of the track with velocity v0 to the right. It reaches the evenly spaced positions x=x1,x=x2... at later times t1, t2...

The question now is, given that the spatial intervals between the various xi are equal, it is clear that the time intervals t10=t1-t0, t21=t2-t1, etc will probably not be. Which ordering would be possible for these intervals?

The answer is t10>t21>t43>t32.

But why? why is t21 longer than t43 and t43 longer t32? And also, why does the velocity increase when the ball goes down the ramp and decrease when the ball goes up the ramp? Possible to explain in terms of kinematics? If cant, possible to explain using Newton 2nd law and energy conservation in detail to me?

Thanks a million!

 

[Doc Al]

But why? why is t21 longer than t43 and t43 longer t32?

Since the distance traveled is the same, you need to compare the average speeds in those intervals.

And also, why does the velocity increase when the ball goes down the ramp and decrease when the ball goes up the ramp?

Because there's a net force due to gravity acting down the ramp. Per Newton's 2nd law, the ball's acceleration is down the ramp. When the ball is going down, it speeds up, since velocity and acceleration are in the same direction; When going up, it slows down.

Possible to explain in terms of kinematics? If cant, possible to explain using Newton 2nd law and energy conservation in detail to me?

I assume you have a textbook. Read it!

This may help: Inclined Planes

 

[rela]

Wow..Thanks... You have clarified some of my doubts. I will read up more to enhance my understanding and also check out the site you've given.

Appreciate your leads once again.

:)