Aim for FivePhysics 1

Translation between representations (TBR)

Satellite orbits and escape

  • Units 2, 3 and 6
  • 12 points
  • About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 25–30 minutes.

The question

A satellite of mass m moves in a circular orbit of radius r around a planet of mass M. The planet is much more massive than the satellite, and the only force on the satellite is the planet's gravitational force. Express your answers in terms of m, M, r, and physical constants such as G, as appropriate.

Suggested time: 28 minutes

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Part (a)

1 point

Describe the free-body diagram of the satellite.

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Part (b)

3 points

Derive expressions for (i) the orbital speed v and (ii) the orbital period T of the satellite.

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Part (c)

2 points

Describe the graph of orbital speed v as a function of orbital radius r that you would draw, from small r to very large r.

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Part (d)

3 points

Thrusters move the satellite from its orbit of radius r into a new circular orbit of radius 2r. For the satellite–planet system, compare (i) the kinetic energy, (ii) the gravitational potential energy, and (iii) the total mechanical energy in the new orbit with those in the original orbit. Use expressions to support each comparison.

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Part (e)

3 points

(i) Derive an expression for the minimum speed the satellite would need at radius r to escape the planet entirely. (ii) Using your energy expressions from part (d), explain why this escape speed is greater than the orbital speed at the same radius.

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