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it's doable if the oval is an ellipse.
the equation for an ellipse can be represented a parametric equation:
x = a *Math.cos(t);
y = b *Math.sin(t);
where a and b are the lengths of the major and minor axes and the slope of the ellipse at point x,y is given by m:
m= - b*b*x/(a*a*y);
that's all the data you need to use from your oval. from there it's just a matter of grinding out the path that your ball will travel using some elementary physics.
Wow, thanks! I will love trying this when I have time!