A uniform rod of length \(2\) m and mass \(16\) kg is thrown vertically from the ground, such that the top of the rod reaches a height of \(h\) (from the ground). At the moment that it reaches its apex, a ball of mass \(3\) kg traveling horizontally hits the top of the rod with a velocity of \(10\) ms\(^{-1}\) and binds to the very end of the rod. The rod now starts to spin. It perfectly completes one full rotation and then hits the ground.

Find \(h\) in meters.

**Details and Assumptions**

- The ball is a point particle.
- The rod has no cross-sectional area.
- There is no air friction.
- A complete rotation indicates that after rotating through an angle, the ball-rod system is vertical again, with the ball at the top.

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