Apply the relationships for orbital motion of satellites and planets, including Kepler's third law T^2 / r^3 = 4 pi^2 / (G M), orbital speed v = sqrt(G M / r), and the energy of an orbit (kinetic, gravitational potential and total)

What this dot point is asking

QCAA wants you to derive orbital relationships from first principles by setting gravitational force equal to centripetal force, apply Kepler's third law $T^2 / r^3 = 4 \pi^2 / (G M)$ to planetary and satellite systems, and compute the kinetic, gravitational potential and total energies of a circular orbit. This dot point appears in IA1 (orbital data tables to interpret), EA Paper 1 multiple choice on Kepler proportions, and EA Paper 2 derivations.

The answer

Orbital speed from gravity equals centripetal force

For a satellite of mass $m$ in a circular orbit of radius $r$ around a central body of mass $M$, the gravitational force supplies all of the centripetal force:

Cancelling $m$ and rearranging:

Important features:

• IMATH_15 does not depend on the satellite mass $m$. A spaceship and a bolt would orbit at the same speed at the same altitude.
• IMATH_17 decreases with increasing $r$. Distant orbits are slower.
• The satellite is in continuous free fall toward the central body, but its tangential velocity keeps it perpetually missing.

Kepler's third law

Substituting $v = 2 \pi r / T$ into $v^2 = G M / r$:

For all bodies orbiting the same central mass, the ratio $T^2 / r^3$ is constant. This is Kepler's third law (originally stated for planets around the Sun, but applicable to any central-body system).

In ratio form, between two satellites of the same central body:

This is the working form for QCAA problems that do not give you $M$ directly.

Try it: Kepler's third law calculator. Enter central mass and radius (or period) to get the other.

Kepler's first and second laws (qualitative)

QCAA may ask you to state these as background.

• First law. Planetary orbits are ellipses with the Sun at one focus. Circular orbits are the special case of zero eccentricity.
• Second law. A line from a planet to the Sun sweeps out equal areas in equal times. Equivalently, a planet moves faster when it is closer to the Sun (consistent with conservation of angular momentum).

Most calculations in QCE Physics use the circular-orbit simplification, but you may need to invoke the second law in IA1 when given an elliptical-orbit stimulus.

Energy of a circular orbit

The kinetic energy of a satellite in a circular orbit of radius $r$ is:

(using $v^2 = G M / r$). The gravitational potential energy, taking zero at infinity, is:

The total mechanical energy is:

Key features:

• IMATH_25 is negative, indicating a bound orbit.
• IMATH_26 for any circular orbit. This is the virial theorem for an inverse-square force.
• To raise a satellite to a higher orbit, you must add energy (move $E_{\text{tot}}$ closer to zero), and the satellite ends up moving more slowly. The work done lifts it against gravity faster than the kinetic energy can be replenished.

Escape velocity

The minimum launch speed from radius $r$ that lets a projectile reach infinity with zero kinetic energy:

From Earth's surface, $v_{\text{esc}} \approx 11.2$ km/s. From low Earth orbit, the orbital speed is about $v_{\text{orbit}} \approx 7.9$ km/s, and the additional $\Delta v$ to escape from there is about $3.3$ km/s.

Try it: Escape velocity calculator and orbital energy calculator.

Worked example: a low Earth orbit

A satellite orbits 400 km above Earth's surface ($r = R_E + h = 6.77 \times 10^6$ m). Orbital speed:

$v = \sqrt{\frac{G M_E}{r}} = \sqrt{\frac{6.67 \times 10^{-11} \times 5.97 \times 10^{24}}{6.77 \times 10^6}} = 7.67 \times 10^3 \text{ m/s}$.

Period:

$T = \frac{2 \pi r}{v} = \frac{2 \pi \times 6.77 \times 10^6}{7670} = 5550 \text{ s} \approx 92 \text{ minutes}$.

This matches the orbital period of the International Space Station.

How this appears in IA1 and IA2

IA1 data test. Expect a satellite or moon table (radii and periods, sometimes a missing column) with a question asking you to verify Kepler's third law or extract $M$ of the central body. Alternatively, a stimulus showing the orbital energies as a function of radius with questions on the virial theorem.

IA2 student experiment. A practical IA2 on orbits is hard to engineer directly, but a frequent design is the simple pendulum used to measure local $g$, then comparing the inferred $G M_E / R_E^2$ against the textbook value. The orbital framework provides the EA-level theory for the Unit 3 justification.

Common traps

Forgetting that $r$ is measured from the centre of the central body, not the surface. Always add the planetary radius for satellites: $r = R_E + h$.

Treating $v$ or $T$ as depending on the satellite mass. Both depend only on $M$ (the central body) and $r$.

Reversing the sign of $E_p$. Gravitational potential energy is negative with zero at infinity. The deeper into the well, the more negative.

Forgetting the factor of 2 in $E_{\text{tot}} = -G M m / (2 r)$. A common slip when working under exam time pressure.

Confusing escape velocity with orbital velocity. $v_{\text{esc}} = \sqrt{2} \, v_{\text{orbit}}$ at the same radius. Escape is from infinity; orbit is a bound circular trajectory.

Using inconsistent units in Kepler's third law. If you mix days and seconds, or kilometres and metres, the constant changes. Always work in SI metres and seconds for QCAA problems.

In one sentence

For a satellite or planet in a circular orbit of radius $r$ around a central mass $M$, gravitational force equals centripetal force, giving $v = \sqrt{G M / r}$, Kepler's third law $T^2 / r^3 = 4 \pi^2 / (G M)$, and a total mechanical energy $E_{\text{tot}} = - G M m / (2 r)$ that is negative for a bound orbit.