Physics HL · Chapter 17: Gravitation
17.4 Total Energy, Escape Speed, and Orbital Decay
Classify trajectories by total energy, derive escape speed from energy arguments, and explain why drag makes low satellites spiral inward while speeding up.
Estimated time: 46 minutes
Total Energy as a Trajectory Classifier
In gravitational motion, total mechanical energy ET = EK + Ep determines path type. Negative ET corresponds to bound motion (elliptic family, including circular). Zero ET marks the escape threshold (parabolic limit). Positive ET corresponds to unbound trajectories (hyperbolic family). This classification is often faster than direct trajectory integration.
E_T = rac{1}{2}mv^2 - rac{GMm}{r},qquad E_{T, ext{circular}} = - rac{GMm}{2r}
Circular-orbit total energy is always negative and half the potential-energy magnitude.
The circular-orbit expression is especially useful when comparing nearby orbit radii. Lower radius means more negative total energy. If a satellite loses energy to drag, it must move to a smaller-radius orbit to remain circularized, and smaller radius implies larger orbit speed from v = sqrt(GM/r).
Deriving Escape Speed from Any Radius
Escape speed is the minimum launch speed that lets an object reach infinity with zero remaining speed. That means total energy at launch must be exactly zero. Setting ET = 0 in the energy expression gives the standard relation. This derivation also shows escape speed is independent of the escaping object's mass.
v_{ ext{esc}} = sqrt{ rac{2GM}{r}} = sqrt{2gr} = sqrt{-2V}
All three forms are equivalent when g and V are evaluated at the same launch radius.
If an object is already in circular orbit, extra speed needed to escape is not simply vesc because orbital speed is already present. The required delta-v depends on burn direction and orbital state, but the energy target remains the same: raise specific total energy to zero or above.
Atmospheric Drag, Inward Spiral, and Speed Increase
A low satellite experiences drag opposite motion, so mechanical energy decreases. Because circular-orbit energy equals -GMm/(2r), lower energy means smaller r. Then v = sqrt(GM/r) rises as radius shrinks. So drag can reduce orbital altitude while increasing speed at each new lower orbit. This feels paradoxical only if energy and speed are not tracked together.
Important
For circular orbits, 'losing energy' does not mean 'slowing down forever'. It means dropping to a lower orbit where the required orbital speed is higher.
Simulation: Escape Threshold and Orbit-Energy Analyzer
Set launch radius and speed, then read specific energy, trajectory class, turning radius (if bound), and residual infinity speed (if unbound).
Gravitation + Orbit Lab
Escape speed
11.186 km/s
Circular speed
7.910 km/s
Specific total energy
-1.45e+7 J/kg
Trajectory class
Bound (returns)
Energy line vs potential curve (radial launch model)
Launch diagnosis
This launch is bound with a radial turning point near 2.74e+4 km.
Energy bookkeeping
Specific kinetic = 4.80e+7 J/kg, specific potential = -6.26e+7 J/kg.
Test Yourself
At a location where g = 4.0 N/kg and r = 9.0 x 10^6 m, enter escape speed in km/s.
Hint: Use v_esc = sqrt(2gr).