Satellite KE, GPE and total energy
For a circular orbit, KE = +GMm/2r, GPE = −GMm/r and TE = KE + GPE = −GMm/2r. So KE = −TE = −½ GPE, and the total energy is always negative, which is what bound means.
KE = +GMm/2r · GPE = −GMm/r · TE = −GMm/2rWhat this actually means
Derive KE rather than memorising it. From GMm/r² = mv²/r you get mv² = GMm/r, so KE = ½mv² = GMm/2r. Show that line and the KE mark is safe even if you misremember the final expression.
GPE is just the definition, −GMm/r. Add them and the halves do the work: −GMm/r + GMm/2r = −GMm/2r. Notice TE is exactly minus KE, and exactly half of GPE.
The negative total energy IS the definition of a bound orbit. You would have to SUPPLY energy, exactly +GMm/2r, to lift the satellite to infinity where all three quantities are zero. A body with TE ≥ 0 is not bound and escapes.
Structured questions chain these: ω = 2π/T, then v = rω, then a = rω², then F = ma, then M from F = GMm/r², then KE = ½mv², then GPE = −GMm/r, then TE. Each answer feeds the next, so an early slip cascades. Carry full precision through and round only at the end.
Sign audit before you write the final line. KE positive, GPE negative and twice the size of KE, TE negative and equal in magnitude to KE. If your three numbers do not obey that pattern, something is wrong.
Worked scale for a 2400 kg geostationary satellite: KE = +1.13 × 10¹⁰ J, GPE = −2.27 × 10¹⁰ J, TE = −1.14 × 10¹⁰ J.
Giving the total energy as positive, or forgetting that GPE is twice the magnitude of KE.
Prove it — watch it be true
- Open the KE, PE and Total energy bars and pick any orbit radius.
- Check the KE bar is exactly half the height of the GPE bar and opposite in sign.
- Confirm the Total bar sits below zero at every radius and equals minus the KE bar.