IB Physics HLTopic 4 — Force FieldsPaper 1 & 2Loses energy, speeds up~15 min read
Drag on Orbits
Everything so far assumed a perfect orbit — no air, no friction, total energy fixed for eternity. But the atmosphere does not stop at a tidy line. It thins, and thins, and below about 600 km a satellite is still ploughing through the last few molecules of it. It loses energy to drag. Its orbit shrinks. And then the strangest thing in this whole topic happens: it goes faster. A friction force that speeds things up. This page explains how that can possibly be true.
📘 What you need to know
Satellites in low orbits (below about 600 km) are slightly affected by viscous drag — air resistance
The air density up there is very low, but not zero
Friction between the satellite and air particles dissipates kinetic energy as thermal energy
So the satellite’s total energy decreases, becoming more negative
Since Etotal = −GMm/2r, a smaller total energy means a smaller orbital radius — the satellite spirals inwards
In the lower orbit, some potential energy is transferred to kinetic energy, so the satellite’s speed increases
The drop in Ep is exactly twice the rise in Ek, so the total energy still falls: ΔEtotal < 0 when ΔEp > ΔEk in magnitude
Faster satellite + denser air → even more drag. The process accelerates, and ends in re-entry
There is no edge to the atmosphere
Below roughly 600 km, a satellite is not in a vacuum. The density of air there is fantastically small — but it isn’t nothing, and the satellite is passing through it at nearly 8 kilometres every second. Collisions with those stray molecules rub against the spacecraft’s surface, and kinetic energy is dissipated into thermal energy. The satellite warms; the air warms; the orbit pays.
Over a single orbit the effect is almost too small to measure. Over months and years, it decides when the satellite comes down.
Drawn steeply for clarity. A real decay takes thousands of orbits — but the shape is exactly this: a slow, tightening spiral, then a very fast finish.
The paradox: losing energy makes it faster
Here is where students, quite reasonably, refuse to believe the physics. Drag is a resistive force. It opposes motion. Surely it must slow the satellite down?
It does — for an instant. But the moment the satellite slows, it no longer has enough speed for its orbit, so it falls a little closer in. And gravity, doing positive work on it as it falls, hands it back more kinetic energy than drag took away. Look at the equations from the last page:
The three energies, once moreEk = + GMm/2rEp = − GMm/rEtotal = − GMm/2r
Now let r get smaller and read off what happens to each one.
Ek = +GMm/2r — smaller r means biggerEk. The satellite speeds up.
Ep = −GMm/r — smaller r means more negativeEp. It falls, by twice as much.
Etotal = −GMm/2r — smaller r means more negative total. Energy has been lost, as drag demands.
Two curves go down, one goes up. The satellite is losing energy overall — and gaining speed while it does so.
The resolution is that drag is not the only force doing work. Drag removes energy; gravity adds kinetic energy as the satellite falls. Gravity wins on kinetic energy, and loses on the total. If it helps, think of a cyclist freewheeling down a hill into a headwind. The wind is slowing her, and she is still accelerating — because the hill is steeper than the wind is strong. A decaying satellite is permanently freewheeling downhill.
Doing the accounting properly
Because Ep changes by twice as much as Ek, and in the opposite direction, the two do not cancel. Write it out:
Why the total still fallsΔEp = −2 ΔEkΔEtotal = ΔEk + ΔEp = ΔEk − 2ΔEk = − ΔEkthe energy dissipated as heat is exactly equal to the gain in kinetic energy
That is the sentence IB wants: ΔEtotal < 0 because the decrease in potential energy is larger than the increase in kinetic energy.
The teal bar and the green bar are the same length. The energy the satellite gained as speed is exactly the energy it lost to heat — both paid for by the collapsing potential energy.
Why it accelerates towards the end
The spiral is not steady. It is a runaway.
Drag removes energy
so r falls
Faster satellite, denser air
so drag grows
Even more energy removed
Each loop of that cycle makes the next loop worse. The satellite descends slowly for years, then very quickly for a few weeks, and burns up in seconds.
Quantity
As the orbit decays
Because
Orbital radius r
Decreases
Energy is lost, so Etotal = −GMm/2r falls
Speed v
Increases
v = √(GM/r), and r is smaller
Kinetic energy
Increases
Ek = +GMm/2r
Potential energy
Decreases (twice as much)
Ep = −GMm/r
Total energy
Decreases
Dissipated as thermal energy
Period T
Decreases
T² ∝ r³
🔥 Answering a drag question
Start with the energy. “Drag dissipates kinetic energy as thermal energy, so Etotal decreases.”
Link total energy to radius.Etotal = −GMm/2r, so a more negative total means a smaller r.
Then get the speed.v = √(GM/r) — smaller r, larger v. Say “it speeds up” out loud, and mean it.
Explain the apparent contradiction.Ep falls by twice what Ek gains, so the total still drops.
Finish with the feedback. Lower orbit → denser air → more drag → faster decay.
WE 1
A satellite of mass 800 kg decays from a circular orbit 400 km above the Earth’s surface to one 350 km above it. Calculate the change in its kinetic energy, its potential energy and its total energy, and state how much energy has been dissipated. (GME = 3.98 × 10¹⁴, RE = 6.37 × 10⁶ m)
Step 1 — the two orbital radiir₁ = 6.77 × 10⁶ m r₂ = 6.72 × 10⁶ mStep 2 — kinetic energy at each radius, Ek = GMm/2rEk1 = 2.3527 × 10¹⁰ J Ek2 = 2.3702 × 10¹⁰ JΔEk = +1.75 × 10⁸ JStep 3 — potential energy changes by twice as much, downwardsΔEp = −2 × ΔEkΔEp = −3.50 × 10⁸ JStep 4 — add them for the totalΔEtotal = (+1.75 − 3.50) × 10⁸ΔEtotal = −1.75 × 10⁸ JStep 5 — where did it go?1.75 × 10⁸ J dissipated as thermal energyThe satellite is now travelling 28 m s⁻¹ faster than before, having lost 175 million joules. Speed up, energy down. Both true.
WE 2
Show that, for a satellite decaying between two circular orbits, the energy dissipated by drag is exactly equal to the gain in kinetic energy.
Step 1 — write the two energiesEk = +GMm/2r Ep = −GMm/r = −2EkStep 2 — so a change in r changes them togetherΔEp = −2 ΔEkStep 3 — add for the totalΔEtotal = ΔEk + ΔEp = ΔEk − 2ΔEk = −ΔEkStep 4 — interpret
The energy lost from the system is |ΔEtotal| = ΔEk, and it has gone to heat.
energy dissipated = gain in kinetic energyA tidy result, and it is why the numbers in WE 1 came out equal. Gravity supplies twice what drag takes; drag takes half of what gravity supplies.
WE 3
A student writes: “Air resistance is a resistive force, so it must slow the satellite down.” Explain what is wrong with this statement, and describe what happens to the satellite over a long period of time.
Step 1 — what drag really does
Drag dissipates kinetic energy as thermal energy, so the satellite’s total energy decreases.
Step 2 — the consequence for the orbit
Since Etotal = −GMm/2r, a more negative total means a smaller r.
Step 3 — and for the speedv = √(GM/r), so a smaller r gives a larger v.
the satellite actually speeds up
Drag is not the only force doing work — gravity does positive work as the satellite falls inwards.
Step 4 — the long term
Lower orbit → denser air → more drag → faster decay.
it spirals inwards and burns up on re-entryThe student is not silly — they are just forgetting that a satellite is not on rails. It is free to fall, and falling is what pays for the extra speed.
💡 Top tips
Always start from energy, not from force. “Drag dissipates KE as thermal energy” is the opening line.
Use Etotal = −GMm/2r to get from “loses energy” to “r decreases”. That is the pivot of the whole argument.
Then use v = √(GM/r) to get from “r decreases” to “speeds up“.
Quote the reason the total still falls: ΔEp is twice ΔEk, and opposite in sign.
Say thermal energy or internal energy, not “heat energy” and never “lost energy”.
Mention the feedback: lower orbit, denser air, more drag. It’s often the final mark.
Remember the threshold: drag matters below roughly 600 km.
⚠ Common mistakes
Saying drag makes the satellite slower. Its speed increases as the orbit decays
Saying the satellite gains energy because it speeds up. Its total energy falls
Thinking Ek and Ep changes cancel. ΔEp is twice the size of ΔEk
Claiming the upper atmosphere is a vacuum. Its density is very low but not zero
Forgetting that gravity also does work on the falling satellite — drag is not the only force in play
Saying energy is “lost”. It is dissipated as thermal energy in the satellite and the air
Assuming the decay is steady. It accelerates, because drag grows as the satellite descends
Quick recap: Below about 600 km the air is thin but not absent, so viscous drag dissipates kinetic energy as thermal energy and the satellite’s total energy falls. Since Etotal = −GMm/2r, the orbital radius shrinks — and because v = √(GM/r), the satellite speeds up as it descends. The total still falls because ΔEp = −2ΔEk: the potential energy drops by twice what the kinetic energy gains. Lower orbit, denser air, more drag — the spiral accelerates, and ends in re-entry.
And that closes Gravitational Fields. Look back at the distance covered: one equation for the force between two masses, turned into a field, then into a potential, then into a landscape of contour lines. From that landscape came Kepler’s three laws, the escape speed, the energy of every satellite ever launched — and finally the reason they all, eventually, come home. Newton wrote the first line of it. You have just read the whole chapter.
Still can’t accept that drag speeds it up?
Book a free meeting and we’ll work through the orbital energy argument until the paradox stops being one.
