Physics • Year 12 • Module 5 • Lesson 12
Energy in Orbits
Lock in the four key formulas, the sign convention for gravitational potential energy, and the physical meaning of negative total energy before tackling calculations.
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, binding energy, escape velocity, centre-to-centre distance, virial theorem, 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 magnitude of the total orbital energy; the minimum energy that must be supplied to move the satellite from its orbit to infinity. | |
| 1.5 | The minimum initial speed an object must have to escape the gravitational field of a body from a given distance. | |
| 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 | A relationship stating that for a bound gravitational orbit, the average KE equals −½ the average total potential energy. | |
| 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 Escape velocity depends on the mass of the escaping object: a heavier spacecraft needs a greater initial speed than a lighter one launched from the same point. 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 Escape velocity equals the orbital velocity at the same radius, because both are derived from the gravitational force balance. 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 · binding · escape · 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 the energy required to free the satellite from its orbit, and equals |Etotal|. The minimum launch speed needed to reach infinity is called the ___________ velocity.
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 escape velocity is √2 times greater than orbital velocity at the same radius.
4.4 What does “binding energy” represent, and how is it calculated from the total orbital energy?
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 · binding energy · escape velocity · orbital radius.
Q1 — Term–definition match
1.1 kinetic energy (orbit) • 1.2 gravitational potential energy • 1.3 total orbital energy • 1.4 binding energy • 1.5 escape velocity • 1.6 centre-to-centre distance • 1.7 virial theorem • 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. Escape velocity ve = √(2GM/r) is independent of the escaping object’s mass m, because m cancels on both sides of the energy equation. A heavy spacecraft and a light probe require the same initial speed from the same launch point (ignoring atmospheric drag).
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 False. Escape velocity is √2 times greater than orbital velocity at the same radius: ve = √(2GM/r) = √2 × vorbital. The escape condition requires Etotal = 0, which needs twice the KE of an orbit at the same radius.
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 / binding / escape.
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 — Why ve = √2 × vorbital
At orbital velocity, KE = ½GMm/r and total energy = −½GMm/r. To escape, the satellite needs Etotal = 0, so its KE must equal GMm/r — exactly twice the orbital KE. Since KE ∝ v², doubling KE means multiplying v by √2. Hence ve = √(2GM/r) = √2 × vorbital.
Q4.4 — Binding energy
Binding energy is the minimum energy that must be added to a satellite to move it from its current orbit to infinity (where E = 0 and the satellite is just free). It equals the magnitude of the total orbital energy: Ebinding = |Etotal| = ½GMm/r. A satellite with a more negative total energy has a larger binding energy and is more tightly bound.
Q5 — Sample concept map
Correct maps should include arrows such as:
- kinetic energy + gravitational potential energy — sums to give → total orbital energy
- total orbital energy — magnitude equals → binding energy
- orbital radius — inversely determines → kinetic energy (KE = ½GMm/r)
- orbital radius — inversely determines → escape velocity (ve = √(2GM/r))
- gravitational potential energy — equals −2 × → kinetic energy
- binding energy — is released when satellite reaches → escape velocity
Award 1 mark per valid labelled arrow with a correct linking phrase (minimum 6 marked).