MHB Acceleration ball rolling on a parabola

[Barioth]

Hi, I'm trying to find the Acceleration of a ball rolling down on a parabola.


If I could find it, then I could integrate it twice and find it's parametric equation given by the time.
How could I find this?
I tried a few things, but nothing that made any sense.
If someone could give me a hint on where to start it would be super!


Thanks for passing by!

 

[Ackbach]

A couple of questions:

1. Do you have the equation for the parabola?

2. Are you ignoring rotational kinetic energy or not?

3. What is the radius of the ball?

 

[Barioth]

Sorry I should have given more information!

1- The equation is given but is variating, so I have to find a solution for its general form.

2- We're ignoring pretty much everything except gravity.

3-Since we're ignoring everything the radius isn't told.

 

[Ackbach]

Sorry I should have given more information!

1- The equation is given but is variating, so I have to find a solution for its general form.

2- We're ignoring pretty much everything except gravity.

3-Since we're ignoring everything the radius isn't told.

That helps, but I still don't know a few things. Is the parabola opening up? Down? What general form are you given? It might be helpful if you could state the original problem verbatim.

 

[Barioth]

I need to code in maple an animation of a ball rolling down multiple parabola and a straight line.

To find the fastest way for a ball to move from Point A to B.
They all start at the Point (0,0) and end at (K,-1)

K is given by the user before computing. I have 2 different parabola that have some difference in their acceleration.

I've managed to evalute r(t) for the straight line, but I can't figure out how to start with the parabola.
If I have $$y=ax^2+bx+c$$ ( in my case C always equal to 0)

edit: I had guess it is opening up also.

The problem is in French and I'm doing my best to translate it, hope it sounds fine!

 

[MarkFL]

It sounds like this problem is related to the famous problem:

Brachistochrone curve - Wikipedia, the free encyclopedia

 

[Barioth]

Nice call MarkFL it is indeed!

I've managed to deal with the Brachistochrone ( Well I've found some equation that I had to change a litle bit to make it work for me...), but the parabola give me problem.

I might try to go find a physics teacher in my old school, I'm having an hard time translating the question.

 

[Ackbach]

The solution to the brachistochrone (from the Greek for "shortest time") is the cycloid. To prove that requires the calculus of variations, and some technical details. Can you fit a cycloid to the points you have?

 

[Barioth]

Hi, I have solved the cycloide already, (I had to do my semester research on cycloide 3 years ago). I can make it fit the point that are given.

I'll put the parabola on the ice and do some reading, I might need some more knowledge
now!

Thanks for Helping Ackbach!