Higher orbit: slower, but more total energy
Moving to a larger orbit makes the satellite slower (v = √(GM/r)) with less kinetic energy, yet its total energy increases (becomes less negative), so work must be done on it: W = GMm/2 (1/r₁ − 1/r₂).
ΔTE = (−GMm/2r₂) − (−GMm/2r₁) = GMm/2 (1/r₁ − 1/r₂) > 0 for r₂ > r₁What this actually means
This is the paradox questions are built on: a higher satellite moves slower but has MORE energy. Both halves are true and they are not in conflict, so say both clearly.
The bookkeeping resolves it. KE = GMm/2r goes down when r goes up. GPE = −GMm/r also goes up, but it is twice as large in magnitude, so it rises by twice as much as KE falls. The net change is an increase.
So raising an orbit costs energy, ΔTE = GMm/2 (1/r₁ − 1/r₂), even though the satellite ends up moving more slowly. Thrusters supply that energy; most of it goes into potential energy, and some kinetic energy is actually given back.
Watch the wording in comparison MCQs. 'Which satellite has more potential energy?' The higher one, always, because its GPE is less negative. 'Which is faster?' The lower one. Students who assume higher means faster get both wrong.
The same reasoning shows why a low satellite is hard to keep low: it is at the bottom of the energy ladder, moving fastest, and any energy it loses pushes it further down and speeds it up again.
Assuming more energy means more speed, when a higher orbit is both slower and higher in total energy.
Prove it — watch it be true
- Open the higher-orbit-slower-but-more-energy panel and drag the orbit radius outwards.
- Watch the speed readout fall and the KE bar shrink at the same time.
- Watch the Total energy bar rise towards zero, showing energy has to be supplied to get there.