Learning outcomes
- Define average orbital speed.
- Recall and use v = 2πr/T.
- Convert orbital periods into seconds.
- Use standard form and appropriate significant figures.
- Interpret the limitations of the circular-orbit model.
2.1 Distance travelled in one orbit
For a circular orbit of radius r, the distance travelled in one complete revolution is the circumference 2πr. If the period T is the time for one complete orbit, average orbital speed is total distance divided by total time.
The syllabus uses an average radius because actual planetary orbits are slightly elliptical. The calculated value is therefore an average speed. The direction of velocity changes continuously, even when the speed is constant, because velocity is a vector.
2.2 The orbital speed equation
The required relationship is v = 2πr/T, where v is average orbital speed, r is the average radius of the orbit and T is the orbital period. In SI units, r is measured in metres, T in seconds and v in metres per second.
Students sometimes use πr², which is the area of a circle, rather than 2πr, which is the circumference. A quick sketch of a circular orbit helps identify that the moving object travels around the edge, not across the area.
2.3 Unit conversion
Orbital periods are often given in days or years. Convert days using 1 day = 24 × 3600 s. Convert years using the number of days stated in the question, commonly 365 days. Keep the conversion visible so that method marks can be awarded even if arithmetic later goes wrong.
Distances may be given in kilometres. Convert kilometres to metres by multiplying by 1000. A common error is to convert the radius but leave the period in days, producing an answer with meaningless units.
2.4 Standard form and scale checks
Planetary speeds and distances are conveniently written in standard form. Multiplication and division of powers of ten should be completed carefully. The Earth’s orbital speed is of the order 10^4 m/s, whereas light speed is of the order 10^8 m/s.
An answer of only a few metres per second for a planet should be rejected as physically unreasonable. Estimation is a powerful way to identify a missing factor of 1000 or an unconverted day.
2.5 Circular motion interpretation
Gravity provides the force directed toward the centre of the orbit. The planet’s velocity is tangential to the orbit, while the acceleration and resultant force point inward. A planet is not moving toward the Sun because its sideways motion continually carries it around the Sun.
The orbital speed equation calculates a scalar average. It does not by itself calculate gravitational force. Avoid introducing formulas beyond the syllabus unless they are supplied in the question.
Worked examples
Earth’s orbital speed
Using r = 1.50 × 10^11 m and T = 365 × 24 × 3600 s, v = 2πr/T ≈ 3.0 × 10^4 m/s.
A satellite orbit
A satellite has orbital radius 7.0 × 10^6 m and period 5.8 × 10^3 s. v = 2π(7.0 × 10^6)/(5.8 × 10^3) ≈ 7.6 × 10^3 m/s.
Practical focus
Investigation or modelling activity
Create a spreadsheet with orbital radius and period for several planets. Convert all quantities to SI units, calculate average orbital speed and check whether speed decreases as distance from the Sun increases. Include formula cells so an incorrect conversion can be located.
Examination guidance
- Write the equation before substituting numbers.
- Use radius, not diameter.
- Convert every time unit to seconds when the requested speed is in m/s.
- Retain extra digits during working and round only the final answer.
- Include the final unit.
Check your understanding
- What distance is travelled in one circular orbit of radius r?
- Why is the result called an average orbital speed?
- What is the SI unit of orbital period?
- What force keeps a planet in orbit?
Answers
- 2πr.
- The real orbit is slightly elliptical, so radius and speed vary; the equation uses an average radius and period.
- The second.
- The Sun’s gravitational attraction.
