Projectile motion of ball 2

A ball tied to the end of a string is in a uniform circular motion at a speed of \(v,\) along the circular path with radius \(r\) centered at point \(O,\) as shown in the above diagram. The height of \(O\) from the ground is \(2r.\) Suppose that, when the angle between the string and the line drawn from \(O\) perpendicularly to the ground is \(45°,\) the string is broken loose and the ball flies as indicated by the green dotted line. What is the highest height, \(h,\) of the ball during the flight?

\(g\) denotes gravitational acceleration.

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