Physics • Year 12 • Module 5 • Lesson 17
Kepler's Laws & Orbital Mechanics
Lock in the vocabulary, the three laws, and the key formulae before tackling harder questions. Every term you secure here is a mark you won't drop in the HSC.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: Kepler's First Law, Kepler's Second Law, Kepler's Third Law, eccentricity, perihelion, aphelion, semi-major axis, Hohmann transfer orbit, angular momentum, centre-to-centre distance. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The statement that planets orbit the Sun in elliptical paths with the Sun at one focus. | |
| 1.2 | The statement that a line joining a planet to the Sun sweeps out equal areas in equal times. | |
| 1.3 | The statement that T² = (4π²/GM)r³ — period squared is proportional to orbital radius cubed. | |
| 1.4 | The dimensionless quantity e = c/a that measures how elongated an ellipse is; e = 0 for a perfect circle. | |
| 1.5 | The point in an orbit where the orbiting body is closest to the Sun (or central body). | |
| 1.6 | The point in an orbit where the orbiting body is furthest from the Sun (or central body). | |
| 1.7 | Half the longest diameter of an ellipse; equals the orbital radius for circular orbits. | |
| 1.8 | The most energy-efficient elliptical path used to transfer a spacecraft between two circular orbits. | |
| 1.9 | The conserved quantity L = mvr whose constancy explains why planets speed up near perihelion. | |
| 1.10 | The distance r = Rplanet + h that must be used in orbital formulae; measured from the centre of the central body. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 Kepler's First Law states that planetary orbits are perfect circles with the Sun at the centre. T / F
2.2 A planet moves fastest at aphelion, because it has covered the greatest distance from the Sun. T / F
2.3 The ratio T²/r³ is the same for all planets orbiting the Sun, because it depends only on the Sun's mass. T / F
2.4 Kepler's laws apply only to planets orbiting the Sun and cannot be used for artificial satellites orbiting Earth. T / F
2.5 Kepler's Second Law is a consequence of the conservation of angular momentum. T / F
2.6 In orbital calculations, r is the altitude h of the satellite above the planet's surface. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
angular momentum · central mass · circular · eccentricity · ellipses · equal areas · gravitational · period
Kepler's First Law states that planets travel in ___________, with the Sun at one focus. The shape of the orbit is described by its ___________, which equals zero for a perfectly ___________ orbit. Kepler's Second Law states that a line joining the planet to the Sun sweeps out ___________ in equal times. This is a direct consequence of the conservation of ___________, because gravity is a central force. Kepler's Third Law relates the orbital ___________ to the orbital radius: T² = (4π²/GM)r³. The constant 4π²/(GM) depends only on the ___________, not on the orbiting body. The Third Law can be derived by setting the ___________ force equal to the centripetal force for a circular orbit.
4. Function recall
Answer each question in 1–2 sentences using precise terms. 8 marks (2 each)
4.1 What physical quantity is conserved to explain why a planet moves faster at perihelion than at aphelion?
4.2 What assumption is made when deriving T² = (4π²/GM)r³ from Newton's law of gravitation?
4.3 Why must you always use the centre-to-centre distance r = R + h (not just the altitude h) in Kepler's Third Law?
4.4 Why is a Hohmann transfer orbit described as the most energy-efficient transfer between two circular orbits?
5. Build a concept map
Draw labelled arrows between the six terms below to show how they connect. Each arrow must carry a linking phrase (e.g. "is derived from", "results in", "depends on"). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: Kepler's Third Law · Newton's Law of Gravitation · orbital period T · central mass M · orbital radius r · centripetal force.
6. Formula recall card
Complete the table by writing the SI units of every variable and a one-line note on when to use each formula. 6 marks
| Name | Formula | Variables & SI units | When to use it |
|---|---|---|---|
| Kepler's Third Law | T² = (4π² / GM) r³ | T in ___; r in ___; M in ___; G in ___ | |
| Orbital speed ratio | vp / va = ra / rp | v in ___; r in ___ | |
| Hohmann transfer semi-major axis | atrans = (r1 + r2) / 2 | a in ___; r1, r2 in ___ | |
| Hohmann transfer time | ttrans = π√(a³ / GM) | t in ___; a in ___; M in ___ | |
| Eccentricity | e = c / a | e is dimensionless; c and a in ___ | |
| Centre-to-centre distance | r = Rplanet + h | r, R, h all in ___ |
Q1 — Term-definition match
1.1 Kepler's First Law • 1.2 Kepler's Second Law • 1.3 Kepler's Third Law • 1.4 eccentricity • 1.5 perihelion • 1.6 aphelion • 1.7 semi-major axis • 1.8 Hohmann transfer orbit • 1.9 angular momentum • 1.10 centre-to-centre distance.
Q2 — True / false with correction
2.1 False. Kepler's First Law states that orbits are ellipses (not circles) with the Sun at one focus (not the centre). Earth's eccentricity e = 0.017, so its orbit is nearly but not perfectly circular.
2.2 False. A planet moves fastest at perihelion (closest approach), not at aphelion. Conservation of angular momentum requires v to increase as r decreases (L = mvr = constant).
2.3 True. T²/r³ = 4π²/(GM⊙), which depends only on the Sun's mass M⊙, so it is the same for all planets orbiting the Sun.
2.4 False. Kepler's laws apply to any orbiting system under gravity — satellites around Earth, moons around planets, exoplanets around other stars, binary star systems, etc.
2.5 True. Gravity is a central force (directed toward the Sun), so there is no torque about the Sun, and angular momentum L = mvr is conserved. Equal areas swept in equal times follows directly.
2.6 False. r is the centre-to-centre distance: r = Rplanet + h. Using altitude h alone ignores the planet's radius and gives a completely wrong answer.
Q3 — Cloze paragraph
In order: ellipses / eccentricity / circular / equal areas / angular momentum / period / central mass / gravitational.
Q4.1 — Physical quantity conserved
Angular momentum L = mvr is conserved. Because gravity is a central force (directed toward the Sun) there is no torque about the Sun, so L cannot change. As r decreases approaching perihelion, v must increase to keep L = mvr constant.
Q4.2 — Assumption in the derivation
The key assumption is that the orbit is circular (eccentricity e = 0). This allows the gravitational force (GMm/r²) to be set equal to the centripetal force (mv²/r = m(2πr/T)²/r). For elliptical orbits the result still holds if r is replaced by the semi-major axis a.
Q4.3 — Why use centre-to-centre distance
Newton's law of gravitation uses the distance between the centres of mass of the two bodies, not the distance from the surface. A satellite at altitude h orbits at r = REarth + h from Earth's centre. Using only h would severely underestimate r (by ~6370 km for a near-Earth orbit) and give a large error in T.
Q4.4 — Why Hohmann transfer is most energy-efficient
A Hohmann transfer uses the minimum number of engine burns (two) and takes the path of minimum Δv by being tangent to both circular orbits. The ellipse connects the two orbits at their periapsis and apoapsis, so no energy is wasted fighting the existing orbital velocity — each burn adds to (or subtracts from) the velocity in the direction of motion.
Q5 — Sample concept map
Correct maps should include arrows such as:
- Newton's Law of Gravitation — provides → centripetal force
- Kepler's Third Law — is derived from → Newton's Law of Gravitation
- centripetal force — equals gravitational force in → Kepler's Third Law derivation
- orbital period T — increases with → orbital radius r
- central mass M — determines the constant in → Kepler's Third Law
- orbital radius r — cubed in → Kepler's Third Law
Award 1 mark per valid labelled arrow (minimum 6, maximum 6 marked).
Q6 — Formula recall card (sample entries)
Kepler's Third Law: T in s, r in m, M in kg, G in N m2 kg−2. Use when finding the period or radius of any circular/near-circular orbit around a known central mass.
Orbital speed ratio: v in m/s, r in m. Use when finding the speed at perihelion or aphelion given the distance ratio and one known speed.
Hohmann semi-major axis: a, r1, r2 all in m. Use as the first step when calculating Hohmann transfer time or period.
Hohmann transfer time: t in s, a in m, M in kg. Use to find how long a spacecraft coasts from one orbit to another along the transfer ellipse (it is half the period of the ellipse).
Eccentricity: dimensionless; c and a in m. Use when a question asks about orbital shape or when distinguishing nearly circular from highly elongated orbits.
Centre-to-centre distance: r, R, h all in m. Use every time you are given an altitude; always add the planet's radius before substituting into any orbital formula.