# Explain why the total mechanical energy of a satellite in a circular orbit around a planet is equal to half the gravitational potential energy at that orbit.

> IB DP Physics Higher Level (2023 syllabus) — D.1 Gravitational fields · Explain · 4 marks

> A satellite of mass m orbits a planet of mass M in a circular orbit of radius r. The gravitational potential energy of the satellite at this orbit is U = −GMm/r.

## Mark scheme (4 marks)

1. For a circular orbit, the gravitational force provides the centripetal force, so GMm/r² = mv²/r, giving the kinetic energy EK = ½mv² = GMm/2r.
2. The gravitational potential energy at radius r is EP = −GMm/r, so the kinetic energy EK = −EP/2 = −½(−GMm/r).
3. The total mechanical energy is E = EK + EP = GMm/2r − GMm/r = −GMm/2r.
4. Therefore E = EP/2 (i.e. the total energy equals half the gravitational potential energy), and the negative value confirms the satellite is in a bound orbit.

## Key terms

- [gravitational potential energy](https://www.gradenine.co.uk/glossary/gravitational-potential-energy)
- [total mechanical energy](https://www.gradenine.co.uk/glossary/total-mechanical-energy)
- [circular orbit](https://www.gradenine.co.uk/glossary/circular-orbit)

## Related

- [Revision notes for IB DP Physics Higher Level (2023 syllabus)](https://www.gradenine.co.uk/learn)
- [How to answer "Explain" questions](https://www.gradenine.co.uk/tools/command-word-cheatsheet)
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Source: [GradeNine](https://www.gradenine.co.uk/q/explain-why-the-total-mechanical-energy-84ff616d) · Published by Druglandscape Ltd.