Which Path is Fastest for a Rolling Sphere: Straight Line or Curve?

[mid_bama]

need help with rolling sphere problem...

Homework Statement

i have to design a track for a rigid sphere moving under the force of gravity. I need to come up with two paths/curves in the vertical plane and prove which one is the fastest. Origin point is (0,0) and ending point is (3,-1.5).

Homework Equations

The Attempt at a Solution

I am thinking that i should prove a straight line isn't as fast as a curve. Any ideas as to how to go about doing this. I believe i could use excel to help run some equations. Thanks for your help...

 

[Delphi51]

Welcome to PF, bama!
Wow, interesting question. I suppose you'll get a faster time by making that sphere speed up early in the run.

The word "prove" in the question makes me wonder if a numeric solution on the spreadsheet will do. Such calculations are never exact and it can be tricky to make them even pretty good. Might be worth checking with whoever assigned it.

Could you get away with a path that consists of two straight lines? That should be fairly easy to do analytically. Even a simple curve like a parabola could be difficult - do you have calculus?

 

[mid_bama]

Thanks Delphi!
I didn't think about two straight lines. She just said it had to follow a path. I just need to find a way to express this in my spreadsheet. She wants us to design a path and then build it in a real world environment.

 

[Delphi51]

Okay, sounds like fun. The two straight lines could cause a bounce in the real world. You'll have to round the join at least. Likely you could model a short circular section in your spreadsheet. Don't forget the rotational energy in your model. I hope there is a prize for the best time!