A ball falls undal gravity from a height of 10m with an initial downward velocity u. It collides with ground, loses halft its energy and then rises back to the same height. Find the initial velocity u.

Explanation

To solve the problem step by step, we will analyze the motion of the ball as it falls, collides with the ground, and then rises back to the original height.

### Step 1: Determine the mechanical energy at the initial height (A)

The ball is dropped from a height of \( h = 10 \, \text{m} \) with an initial downward velocity \( u \). The total mechanical energy at point A (just before it starts falling) is given by the sum of its kinetic energy and potential energy.

\[
E_A = KE_A + PE_A
...