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Dumb_as_a_Duck
Below is an attachment containing my proof of how the total mechanical energy for a freely falling ball remains constant even at different positions.
There's a problem with page 3, where I've tried to prove that M.E. of the ball at position C when it is on the surface of the ground. My textbook says that at this position, intitial velocity 'u' is zero. I don't find that possible, as just at postion B it was moving with a velocity (when squared) of 2gx. Could you see if my version of deriving is correct? (that's on page 3. the other two have been attached for convenience/reference).
If the projectile has zero velocity at point C and is resting on the ground, it possess no mechanical energy. This obviously cannot be the case since energy is conserved as the projectile falls. Your proof looks correct.