If you roll a ball down the stairs, does it get to the flag faster than if you just roll it along flat ground?

roll, slide, whatever.

Also, kinetic friction exists, and will slow the ball down

No, it won't... A ideal ball rolling without slipping will not have an kinetic friction applied to it.

youtube.com/watch?v=_GJujClGYJQ

thread over, go home, kids.

This is a rather different scenario because the track is arranged in such a way that the graviational force will have a component in the horizontal direction.

>if there is friction

The ball at the top of the stairs has the potential energy of height to convert into forward kinetic motion. Given an equal push such that the lower ball will reach the flag, the other ball should "cover" the horizontal distance faster, though it might bounce over the flag (does that still count?).

The stairs.
If the ball is already in motion, nothing is opposing the momentum of the ball in both cases except friction.
case 1: no bounce. This would then be the same as an incline: the stairs is adding energy to the ball from the vertical component that is conserved, meaning the ball is moving faster at the bottom of the stairs.
case 2. Bounce. The forward momentum either stays the same, but has less friction since it is not touching the ground, or the ball stretches in favor of its forward motion, and the elasticity propels the ball forward.

The only degenerate case would be if the stairs were slanted backwards and deflected the ball backwards.

Either case, conservation of energy says the stairs win.

I don't understand what you are saying. It doesn't matter if the ground has kinetic friction coefficient, an ideal rolling object is not affected by kinetic friction. Google it, its basic.

This is wrong because it assumes that the gravitational potential energy conversion to kinetic energy is such that the horizontal velocity is affected. If the ground was near perfectly inelastic then the ball would simply strike the steps, not bounce at all, and a loud noise or heat wave in the ground material would be released corresponding to the energy difference. If the ground was less than perfectly inelastic, then the ball would bounce somewhat as you say, but in neither case does the ball roll faster.

This track is arranged so that the normal force from the track applies horizontal acceleration on the ball which doesnt happen in this problem. Thats what identifies it as a simple physics problem rather than a hamilton's principle problem.

>case 1: no bounce. This would then be the same as an incline: the stairs is adding energy to the ball from the vertical component that is conserved, meaning the ball is moving faster at the bottom of the stairs.
Incorrect, see above. If the ball does not bounce the energy is lost through the medium.

>case 2. Bounce. The forward momentum either stays the same, but has less friction since it is not touching the ground, or the ball stretches in favor of its forward motion, and the elasticity propels the ball forward.
I did not consider if the ball is elastic, or how that would affect anything, but your note of friction is incorrect. The ball is not slowed down by the static friction of the ground since it does no virtual work.

are you literally retarded? nowhere does OP mention the ball is ideal, and kinetic/static friction coefficients are a RELATIVE term, ie: it is uniquely determined from the contact of two different objects, so if a friction coefficient exists, it means neither the ball nor the ground are ideal.

Look at it this way fellas: Lets say this ball was rolling without slipping then dropped off a cliff onto a perfectly elastic ground such that it continues its forward momentum but now oscillates up and down repeatedly. The static friction does no work on the ball, so if the ball was faster in the horizontal direction while in the air, then it would be gaining energy.

No thats completely wrong. An ideal ball is simply one so that a lone atom of the ball contacts the ground at any given instant. An ideal ground, which I did not even mention, just implies perfect rigidity. It is not required that these be violated for friction to exist. Electrostatic forces between the lone atom and the ground dependent on some tricky potential is suffecient to ensure static friction.

can you seriously just kill yourself

>friction doesnt exist
>no, friction does exist
>hurr durr electrodynamics

No I didn't say friction exists and doesnt exist. You are being pedantic and stupid. The long and short of it is that kinetic friction does not apply on an ideal ball since the lone atom in contact with the ground is in contact only for an instant. The lone atom does not "slide" on the ground. Similarily the static friction does no work because it applies on the lone contact atom of the ball only for an instant. For a more rigorous outline of these principles google D'Alembert's principle.

I don't think you understand how rolling works..

Imagine a square wheel that you push foward, not by sliding it, but by throwing the weel over its own corner each time. You should notice that the corners don't move when they are on the floor!! Now try a pentagon, now a hexagon, and go on until infinity. Now you have a circle.
The same goes for the circle - only one point of the circle touches the floor and it DOESNT MOVE relative to floor because the spinning movement EXACTLY counters the speed of the center of the wheel forward.

Obviously the same goes for a ball.

this is the only right answer in this thread

That would only be true if the ball fell with it's center of mass just behind the next step. Which would depend on it's speed and the dimensions of the stairs.

probably

when it hits a corner the equal opposite force is at the surface normal which converts some of the gravitational energy of the ball to kinetic energy

I don't think anything in the problem suggests it will hit a corner besides the bad art, but reading your post I realized if one wanted to be really particular if you do not consider the ball as a particle in the classical sense that when it falls from one step to another there will be a very brief moment in which the corner applies a normal force that has a horizontal component, accelerating the ball forward. However, I want to reiterate that even in such a case, the acceleration of the ball forward has nothing to do with friction, as many individuals here have mistakenly assumed.

not sure if you are pretending to be a smart ass or if you are a smartass

whether or not it hits a corner doesn't only depend on the shape of the stairs it also depends on the velocity it is rolled at

I really don't think it matters how fast its going. If the corner is at a perfect 90 degrees the corner will contact the ball at some point besides its absolute bottom as the ball moves past the corner. Maybe in the real world if we imagine a ball shooting off a cliff we dont think of the ball touching the corner at all but gravity still exists and so in the extremely small time interval the corner is still under the ball as it zooms off the stair, gravity will have pulled the ball downward so some point close to the ball's bottom is in contact with the ball.

Yes. I tried it at home and the stairs are faster

empirical knowledge > theoretical

If only we could reproduce his particular brand of radioactive shag carpeted stairs in the lab.

What if it looked like this? Intuitively you'd think that they would reach the flag at the same speed, still. Is there something about the multiple steps that makes the ball accelerate?

If the balls are rolling slow enough the top ball might get a little torque when it's center of mass is right after the step and it's end is still on.

This: is correct.
In fact it is not a question of might. It will happen, although this is a very subtle quality, and the acceleration of the ball from this will be small on an extreme order. If however, a slightly different set up was created so that the ball goes through the same motion without ever "going over" a corner, say if it was held up by platforms on pistons and the pistons dropped the platform the ball was lying on at very high speed, mimicing the staircase set up, then I would say the ball gets there at the same time.

so basically the more stairs there are the faster the ball would get there?

Well if you think about it, the more stairs there are the smaller they have to be if the initial and final points don't change. As the number of stairs approaches infinity they become a smooth ramp for the ball to roll down which would make it go faster than the ball on the ground. This relationship should hold with even a few stairs, although whether or not the change is significant with only a small number of stairs is difficult to say without testing it.

Yea I would say so. I would also say that with a fixed number of stairs, the faster the ball is rolled then the less acceleration it will receive from the corner due to less contact time at a less advantageous angle. For example if a ball was rolled very slowly it is obvious how the ball comes into contact with the corner as it rolls over it, and if a ball is going extremely fast over the corner one would not immediately think that it may contact the corner at all (from a point besides the bottom of the ball) but in fact it will just at a point very close to the bottom of the ball, hence the small horizontal component. That being said if the ball is going too fast there is an issue of it flying off and scraping a corner some steps down, which would be a much more substantial acceleration resultant.

:)

yes

t. brachistocrone

...

The brachistocrone problem is dependent on the particle moving on a fixed path. If you want a newtonian argument for how a cycloid is the best path its because the path is such that normal force accelerates the ball forward, but that can't happen here, at least not in the way you are thinking.

Of course there is some leeway here and someone already pointed out that the ball could be going so fast it just flies off the handle and never even hits the steps but I think it is sensible to restrict the speed and the dimensions of the stairs so that the ball always lands in the middle or so of the subsequent stair, not on the edge or something.

>Assuming gravity
nice meme

depends, if the stair ball always bounces down the flat horizontal part of the stairs, horizonatal velocity will remain constant (as long as there isn't any friction). If the stair ball hits the very corner of the stairs, the impact will propel the ball forward making it finish before the flat surface ball.

seems right but as already mentioned many times here the ball will accelerate as it goes over the corner regardless whether or not it "lands" on the corner, and a discussion of friction is irrelevant (friction must exist for something to roll, but this friction does not slow the ball down)

And here I thought the portal problem discussions were fucking stupid.