Rearrange the orbital speed equation for period and radius
Find T or r from v = 2πr/T while retaining the model and units.
Section P6.1.1 · The Earth
Mapped to the named 0654 section using teaching from the corresponding single-science course. A 0654-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Rearrange a three-variable equation.
- Recognise the centre-to-surface radius.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which balance follows by multiplying v = 2πr/T by T?
What you will learn
- Find T or r from v = 2πr/T while retaining the model and units.
- Rearrange the speed equation and distinguish calculated radius from surface altitude.
Keep the multiplication together
Starting from v = 2πr/T, multiply both sides by T to obtain vT = 2πr. Divide by v for T = 2πr/v, or divide by 2π for r = vT/(2π). Writing this intermediate balance makes it harder to put a period or π in the wrong place.
A period has units of time: metres divided by metres per second gives seconds. A radius has units of distance: metres per second multiplied by seconds gives metres, with 2π dimensionless. Units can expose a rearrangement error before a numerical answer is accepted.
Interpret the radius you obtain
The calculated radius is the centre-to-orbit distance in the supplied circular or average-radius model. To find an altitude, subtract a separately supplied planetary radius. Do not call the entire orbital radius an altitude or assume a planetary radius when none is given.
A rearranged equation does not establish that the supplied orbit is physically possible around a particular body. It describes distance, path and period consistently in the model. Strongly elliptical orbits still require care about average radius and average versus local speed.
Sources for this lesson
Pause and explain
Which expression gives orbital radius from average speed and period?
Worked example
A circular model has r = 8.0 × 10⁶ m and average v = 8000 m/s. Find T.
Show the worked solution
- Rearrange to T = 2πr/v.
- Substitute 2π × 8.0 × 10⁶ ÷ 8000.
- Calculate 6283.19 s.
Answer Approximately 6.28 × 10³ s, or 105 minutes.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A supplied circular orbit has r = 9.0 × 10⁶ m and v = 7500 m/s. Calculate T.
Give me a hint
Period is circumference divided by the stated average speed.
Compare my reasoning
- Use T = 2πr/v.
- Calculate 2π × 9.0 × 10⁶ ÷ 7500.
- Obtain 7539.82 s.
Approximately 7.54 × 10³ s.
Look for these in your work
- Rearrange before substitution.
- Give the period in seconds.
A supplied circular model has v = 6000 m/s and T = 8000 s. Calculate r.
Give me a hint
Speed multiplied by period gives the complete path length.
Compare my reasoning
- Use r = vT/(2π).
- Calculate 6000 × 8000 ÷ (2π).
- Obtain 7639437 m.
Approximately 7.64 × 10⁶ m.
Look for these in your work
- Keep 2π in the denominator.
- State centre-referenced radius.
For that same orbit, a planet’s supplied radius is 6.0 × 10⁶ m. Estimate the satellite altitude.
Give me a hint
Use the calculated orbit radius and subtract the planet’s radius.
Compare my reasoning
- Keep the unrounded orbit radius 7639437 m.
- Subtract 6000000 m.
- Round 1639437 m to the required precision.
Approximately 1.64 × 10⁶ m above the surface.
Look for these in your work
- Subtract rather than add at this stage.
- Distinguish altitude from orbit radius.
Common mistakes
- Do not invert the circumference-to-speed ratio.
- Calculated radius is not automatically altitude.
- Check that period answers have time units.
What can you explain now?
Which expression gives orbital radius from average speed and period?
Compare with the explanation
r = vT/(2π)
Multiplying speed by period gives the full path length, then dividing by 2π gives its circular radius.
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Make it stick
Rearrange the speed equation and distinguish calculated radius from surface altitude.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.