Gravitational Potential Energy U

U = −GMm/r (in joules, J)

Key properties:

• Always negative for finite r
• U = 0 at r → ∞ (reference point)
• Increases (becomes less negative) as r increases
• Conservative: work done is path-independent

At Earth's surface: U = −GMm/R = −mgR

Near-surface approximation: For small height h above surface: $\Delta U$ = mgh (familiar near-surface formula)

This follows from: $\Delta U$ = GMm(1/R − 1/(R+h)) ≈ GMm·h/$R^{2}$ = mgh for h ≪ R.

Gravitational Potential V

V = −GM/r (in J $kg^{-1}$)

V is U per unit mass: U = mV

Properties:

• Always negative
• V = 0 at infinity
• Gravitational field: g = −dV/dr

At Earth's surface: V = −GM/R = −gR

Escape Velocity from Energy Conservation

Setting total energy = 0 at infinity: ½mv_$e^{2}$ + (−GMm/R) = 0 + 0 v_e = √(2GM/R) = √(2gR)

This shows that escape velocity equals √(2×|V_surface|) / √(1 kg test mass) — or more simply, the kinetic energy needed equals the magnitude of the surface PE.

Binding Energy

Binding energy = energy needed to free the satellite from its orbit:

E_bind = −E = GMm/(2r) = KE

Binding energy equals KE (magnitude of total energy). A satellite needs this much extra energy to escape from its orbit.

Work Done Against Gravity

Moving mass m from r_{1} to r_{2} (r_{2} > r_{1}):

W_against_gravity = $\Delta U$ = GMm(1/r_{1} − 1/r_{2}) > 0

W_by_gravity = −$\Delta U$ = GMm(1/r_{2} − 1/r_{1}) < 0