The ball's gravitational potential energy as a function of its height after release. The ball's kinetic energy as a function of its height after release. The ball's total mechanical energy as a function of its height after release.

a) U = - G m₁m₂ / r, b) K = ½ m (v₀² + 2gy) ² or K = 2 mg² y² c) Em = m g y (2 g y + 1)

Explanation:

Let's write the functions that are requested

a) Gravitational power energy

U = - dF / dr

F = G m₁ m₂ / r²

U = - G m₁m₂ / r

r is the height

b) The scientific enrgia

K = ½ m v²

Cinematic

v² = v₀² + 2 g y

K = ½ m (v₀² + 2gy) ²

If the initial velocity is zero, the ball is released

K = ½ m 4 g² y²

K = 2 mg² y²

c) Mechanical energy

Em = K + U

Em = 2 m g² y² + m g y

Em = m g y (2 g y + 1)