Aim for FivePhysics C: Mechanics

Mathematical routines (MR)

Satellite in orbit around a small planet

  • Units 2, 3 and 6
  • 10 points
  • About 22 minutes

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

A multipart problem where you use math, including calculus, to analyze a physical situation. You draw or sketch a representation such as a free-body diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and then make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 20–25 minutes. This question type started in May 2025, so Physics C: Mechanics free-response questions from 2024 and earlier are built differently.

The question

A satellite of mass m = 500 kg is in a circular orbit around a planet of mass M=6.4×1023M = 6.4 \times 10^{23} kg and radius R=3.4×106R = 3.4 \times 10^6 m. The radius of the orbit, measured from the planet's center, is r = 2R. The planet has no atmosphere, and the gravitational effects of other objects can be ignored. Use G=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2.

Suggested time: 22 minutes

Your answers are saved in this browser as you type.

Something wrong with this question?

What's wrong?

Please don't include personal details.

Part (a)

1 point

Describe the free-body diagram of the satellite in its orbit. List each force, name the object that exerts it, and give its direction.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (b)

2 points

Derive an expression for the orbital speed v of the satellite in terms of G, M, and r. Then calculate its value.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (c)

2 points

Calculate the period of the satellite's orbit.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (d)

3 points

Derive an expression for the minimum additional energy that must be given to the satellite (for example, by a brief rocket burn) for it to escape the planet's gravity completely, starting from its circular orbit. Express your answer in terms of G, M, m, and r. Then calculate its value.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (e)

2 points

Mission controllers instead want to move the satellite to a new circular orbit of radius 3R. A student says: “The satellite moves more slowly in the bigger orbit, so it will have less mechanical energy there. That means moving it out to the bigger orbit releases energy instead of needing energy.” Do you agree or disagree with the student? Justify your answer.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Checking scoring…

Scoring it yourself shows you the rubric, examples and a model answer. Try writing your answer first.