Acceleration ball rolling on a parabola

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SUMMARY

The discussion focuses on calculating the acceleration of a ball rolling down a parabola, specifically in the context of finding a parametric equation using integration. The user is working with a variable parabola equation of the form $$y=ax^2+bx+c$$, where C is always zero, and is primarily concerned with gravitational effects while ignoring rotational kinetic energy. The problem is related to the Brachistochrone curve, which is known for determining the fastest path between two points, and the user has experience with cycloids but struggles with the parabolic case.

PREREQUISITES
  • Understanding of calculus, specifically integration and parametric equations.
  • Familiarity with the Brachistochrone problem and cycloid curves.
  • Basic knowledge of physics concepts related to motion and gravity.
  • Experience with Maple software for coding animations and simulations.
NEXT STEPS
  • Research the calculus of variations to understand the derivation of the Brachistochrone curve.
  • Learn about the dynamics of rolling motion, including the effects of rotational inertia.
  • Explore the implementation of parametric equations in Maple for simulating motion.
  • Investigate different forms of parabolic equations and their applications in physics problems.
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Students and researchers in physics and mathematics, particularly those interested in motion dynamics, calculus applications, and simulation programming using Maple.

Barioth
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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!
 
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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?
 
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.
 
Barioth said:
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.
 
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!
 
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.
 
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?
 
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!
 

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