Orbital Energy & Escape Velocity — Quick Summary

Mr Toogood's Physics · Gravitational fields

AQA 3.7.2.4
E_k = GMm/2r
Kinetic energy
E_p = −GMm/r
Potential energy
E_total = −GMm/2r
Total (binding) energy
v_esc = √(2GM/r)
Escape velocity

KE & PE of an orbiting satellite

Kinetic energy, potential energy, and total energy plotted against orbital radius

Eₖ (positive) is always exactly half the magnitude of Eₚ (negative).

Substituting v²=GM/r into E_k=½mv² gives E_k=GMm/2r. From gravitational potential (zero at infinity): E_p=−GMm/r.

Both share the same GMm/r term — E_k is positive and exactly half the size of E_p. Not a coincidence; it falls straight out of the same force-balance condition that gave us orbital speed.

Total energy (and why it's negative)

E_total = E_k+E_p = GMm/2r − GMm/r = −GMm/2r

The negative sign means the satellite is bound to Earth — it doesn't have enough energy to coast to infinity and stop there.

Binding energy: |E_total| is the energy needed to raise total energy from its current negative value up to zero — exactly the condition for just escaping.
Common slip: don't "fix" a negative total energy — it's the correct answer for a bound orbit. Only a positive total energy should raise a flag (it means the object isn't actually orbiting).

Escape velocity

An object launched from Earth's surface escaping the gravitational field by exceeding the escape velocity

Escape velocity: the minimum launch speed that (just) reaches infinity with zero speed left.

The minimum launch speed for which an object reaches infinite distance with, in the limit, zero speed remaining. It's not a particular orbit's speed, and the object keeps decelerating the whole way — gravity still does negative work on it throughout.

Deriving escape velocity

The boundary between bound and escaping is total energy = 0:

½mv_esc² − GMm/r = 0  →  v_esc = √(2GM/r)

m cancels — escape velocity is independent of the escaping object's own mass. Comparing with orbital speed: v_esc = √2 × v_orbit at the same r — about 1.41×, not double.

Common slip: r is measured from the centre of the body being escaped — for Earth, that's ~6.37×10⁶ m, not the remaining altitude to travel.

LEO vs. GEO: choosing an orbit

Diagram comparing the surface coverage of satellites at different orbital altitudes

Higher altitude sees more of the Earth's surface, but costs far more energy to reach.

LEO (a few hundred–2000 km, ~90 min period): cheaper to reach (less PE to climb, less total energy to raise), gives a stronger signal (closer), but sees only a small patch and sweeps overhead in minutes.

GEO (~36,000 km, 24 h period): stays fixed over one point permanently, at the cost of a longer, weaker signal path and a much higher energy cost to get there.

Real-world applications

Geostationary communications satellites were first proposed by Arthur C. Clarke (1945), over a decade before any nation reached orbit. The first artificial satellite, Sputnik 1 (1957), was in a low orbit — GEO took years longer to achieve given its much greater energy demands. Modern satellites occupy roughly three shells (LEO / MEO / GEO), matched to their job: navigation, communications, imaging, weather.

Worked-example shortcut: the extra KE needed to escape from a circular orbit always equals the size of that orbit's total (binding) energy — no need to separately find v_esc and subtract the current KE.

Exam essentials

Key equations

  • E_k=GMm/2r   E_p=−GMm/r
  • E_total=−GMm/2r
  • v_esc=√(2GM/r)=√2·v_orbit

Reading total energy

  • Negative = bound orbit (correct!).
  • Zero = just escaping.
  • Positive = not in orbit — escaping with energy to spare.

Common slips

  • Don't "correct" a negative total energy — it's physically right.
  • r is from the body's centre, not the surface or remaining altitude.
  • Extra KE to escape a circular orbit = |E_total| of that orbit — a useful shortcut.