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)

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