Derivation of escape velocity formula .
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Escape velocity equation
The Law of Conservation of Energy states that the total energy of a closed system remains constant. In this case, the closed system consists of the two objects with the gravitational force between them and no outside energy or force affecting either object.
Thus the total final energy—potential energy plus kinetic energy—must equal the total initial energy:
TE = TE∞
KE + PE = 0
Substitute values:
mve2/2 − GMm/R = 0
Add GMm/R to both sides of equation:
mve2/2 = GMm/R
Solve for ve2:
ve2 = 2GM/R
Take the square root of each expression to get:
ve = ± √(2GM/R)
Considering our gravitational convention for direction, ve is upward or away from the other object andis thus negative:
ve = − √(2GM/R)
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