Physics • Year 12 • Module 5 • Lesson 12
Energy in Orbits
Lock in the circular-orbit energy formulas, sign conventions, graph trends and energy changes between circular orbits.
1. Term–definition match
Each definition below describes a key concept from this lesson. Write the correct term from the word list in the right-hand column. Word list: kinetic energy (orbit), gravitational potential energy, total orbital energy, energy change, higher circular orbit, centre-to-centre distance, circular energy relation, bound orbit. 8 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The positive energy associated with a satellite’s orbital motion; equals ½GMm/r for a circular orbit. | |
| 1.2 | Energy of position in a gravitational field; defined as zero at infinite separation and negative for all finite distances. | |
| 1.3 | The sum KE + U for an orbiting satellite; always negative for a gravitationally bound system. | |
| 1.4 | The signed difference Ef − Ei between two specified orbital states. | |
| 1.5 | A final circular state with larger r, smaller KE, and greater (less-negative) U and total energy. | |
| 1.6 | The distance measured from the centre of the central body to the centre of the orbiting body; always larger than the central body’s radius. | |
| 1.7 | For a circular orbit, KE = −U/2 and total energy = U/2 = −KE. | |
| 1.8 | A system in which the total mechanical energy is negative; the orbiting body cannot reach infinity without an external energy input. |
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 The gravitational potential energy of a satellite in circular orbit is positive, because the satellite is above the ground. T / F
2.2 Immediately after the first prograde transfer burn, the spacecraft is already in the higher circular orbit. T / F
2.3 A satellite in a larger (higher) circular orbit moves faster than one in a smaller (lower) circular orbit around the same planet. T / F
2.4 The total orbital energy of a bound satellite is always negative, meaning the satellite cannot spontaneously escape to infinity. T / F
2.5 A higher final circular orbit has greater total energy but smaller kinetic energy than a lower circular orbit. T / F
2.6 Doubling the orbital radius of a satellite makes its total mechanical energy less negative (halves the magnitude), so the satellite is less tightly bound. 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:
negative · infinity · kinetic · total · higher · cancels · half · attractive
In orbital mechanics, gravitational potential energy is defined as zero at ___________, so all finite orbits have ___________ potential energy. This convention reflects the ___________ nature of gravity: work must be done against gravity to remove a satellite to infinity. The ___________ energy of a circular orbit equals ½GMm/r, derived by substituting the orbital velocity into the KE formula. When KE and U are added, the total orbital energy equals −___________ the magnitude of U, because the ½ factor in KE only partially ___________ the −1 factor in U. The ___________ energy is KE + U and is negative for the circular orbit. A ___________ final circular orbit has less KE but greater total energy.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 Why is gravitational potential energy defined as zero at infinity rather than at Earth’s surface?
4.2 What is the physical meaning of the statement “the total orbital energy is negative”?
4.3 Explain why adding energy can produce a lower speed in the final higher circular orbit.
4.4 Why must the instantaneous transfer state be distinguished from the final circular orbit?
5. Connect the concepts
Draw labelled arrows between the six terms below to show how they are related. Each arrow must carry a linking phrase (e.g. “equals −½ times”, “is derived from”, “must equal zero at”). Aim for at least 6 labelled arrows. 6 marks (1 per valid arrow)
Supplied terms: kinetic energy · gravitational potential energy · total orbital energy · energy change · transfer trajectory · orbital radius.
Q1, Term–definition match
1.1 kinetic energy (orbit) • 1.2 gravitational potential energy • 1.3 total orbital energy • 1.4 energy change • 1.5 higher circular orbit • 1.6 centre-to-centre distance • 1.7 circular energy relation • 1.8 bound orbit.
Q2, True / false with correction
2.1 False. Gravitational potential energy is always negative for finite r, with the zero reference at infinity. U = −GMm/r. The fact the satellite is above ground is irrelevant to the sign of U in the orbital mechanics convention.
2.2 False. The first burn changes the spacecraft’s velocity at one point and begins a transfer trajectory. A later burn at the destination radius is required to circularise the orbit.
2.3 False. A satellite in a larger orbit moves more slowly. Orbital speed v = √(GM/r) decreases as r increases. Higher-altitude satellites are slower.
2.4 True.
2.5 True. For circular orbits, increasing r decreases KE = GMm/(2r) while total energy −GMm/(2r) increases toward zero.
2.6 True. Etotal = −½GMm/r ∝ −1/r. Doubling r halves the magnitude, making Etotal less negative, the satellite is less tightly bound and closer to the escape threshold.
Q3, Cloze paragraph
In order: infinity / negative / attractive / kinetic / half / cancels / total / higher.
Q4.1, Why U = 0 at infinity
Setting U = 0 at infinity is the only convention that works cleanly for orbital mechanics. Gravity approaches zero at infinite separation, so this is the natural reference point. It ensures the total energy formula Etotal = −½GMm/r and the virial theorem hold consistently for all orbits, regardless of planet size or orbit radius.
Q4.2, Meaning of negative total orbital energy
Negative total energy means the satellite is gravitationally bound: it does not have enough mechanical energy to reach infinity (where E = 0). The satellite is trapped in the gravitational well of the planet. To free it, external energy equal to |Etotal| (the binding energy) must be supplied.
Q4.3, Added energy but lower final speed
A higher final circular orbit has greater, less-negative total energy because its gravitational potential energy has increased. However, circular speed and KE decrease with radius. The gain in U is larger than the decrease in KE, so total energy rises while final speed falls.
Q4.4, Transfer state and final circle
An impulsive burn changes velocity at one point, so the craft initially follows a non-circular transfer trajectory. The circular formulas at the destination radius describe the final state only after a second burn circularises the orbit.
Q5, Sample concept map
Correct maps should include arrows such as:
- kinetic energy + gravitational potential energysums to give → total orbital energy
- initial and final total orbital energydifference gives → energy change
- orbital radiusinversely determines → kinetic energy (KE = ½GMm/r)
- first burnbegins → transfer trajectory
- gravitational potential energyequals −2 × → kinetic energy
- transfer trajectoryrequires a later burn to reach → final circular orbit
Award 1 mark per valid labelled arrow with a correct linking phrase (minimum 6 marked).