Parabowlic

A smooth friction-less bowl is in the form of a paraboloid whose intersection with the x-y plane is a parabola given by \(y=x^{2}\).

A small spherical marble located at the inner surface of the bowl, where the height is \(h=4\) meters, is given an initial velocity \(v_{0}=\sqrt{12g}\). This velocity is tangent to the circular boundary of the horizontal cross-section of the bowl at the marble's position, and it is also parallel to the plane of that horizontal cross-section.

What is the ratio of the marble's highest possible height to its original height? Submit the answer correct to one decimal place.

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