D.1 · IB Physics
Orbital energy of a satellite
Change the radius of a circular orbit around Earth and watch the speed, period and the kinetic, potential and total energies change, then find the energy needed to raise the satellite to a higher orbit.
v = √(GM/r)
T = 2πr / v
Ek = GMm / (2r)
Ep = −GMm / r
E = Ek + Ep = −GMm / (2r)
ΔE = (GMm/2)(1/r₁ − 1/r₂)
G = 6.67×10⁻¹¹ N m² kg⁻², M = 5.97×10²⁴ kg
- Altitude
- Speed v
- Period T
- Ek
- Ep
- Total E
- ΔE (r₁ → r₂)
Try this
- Set r₁ = 7000 km, note Ek, then double r₁ to 14 000 km. Check that Ek halves and that Ep = 2E and E = −Ek at every radius.
- Choose the geostationary preset and confirm the period is close to 24 h. How does T scale when you double r?
- Keep r₁ and r₂ fixed and double the mass: what happens to ΔE? Then compare ΔE for 7000 → 14 000 km with 14 000 → 21 000 km.
Common error: raising a satellite to a higher orbit needs energy, yet its kinetic energy falls; the potential energy rises by twice that fall, so the total energy increases by ΔE = (GMm/2)(1/r₁ − 1/r₂).