Kinetic Energy GmM(R^2 - r^2)/(2R^3) and Speed (G M (R^2 - r^2)/R^3)^(1/2)as a function of radius r.

Here we integrate force times distance on an unimpeded fall from the earth's surface at radius R to a smaller radius r...

∫_R^r (-m (G M r )/R^3) r (* integrate Force time distance going in *)

(G m M (-r^2 + R^2))/(2 R^3)

The work-energy theorem thus allows us to calculate the kinetic energy after that fall.

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K[r_, R_, G_, M_, m_] := (G m M (R^2 - r^2))/(2 R^3) (* Viola !   Kinetic Energy *)

Knowing kinetic energy allows us to calculate the speed after a fall from radius R to radius r...

Solve[1/2m v^2 == K[r, R, G, M, m], v] (* Solve for Speed *)

{{v→ -(-G M r^2 + G M R^2)/R^3^(1/2)}, {v→ (-G M r^2 + G M R^2)/R^3^(1/2)}}

Thus we can define a function which gives speed as a function of r on a fall from the surface at radius R...

v[r_, R_, G_, M_] := (G M (R^2 - r^2)/R^3)^(1/2) (* speed as a fn of radius r *)

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