Motion in the Universe » Orbital Speed
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 8.6
- Use orbital speed = (2 × π × orbital radius) / time period
- Explain where the equation comes from
- Find the orbital radius, including a satellite above the Earth's surface
- Convert km, hours and days to m and s, and rearrange for T and r
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Unlock This CourseWhere the equation comes from
You already know that speed = distance ÷ time (see Speed, Distance and Time). An object in a nearly circular orbit travels once round a circle. So we only need two things: the distance round the circle and the time it takes.
- The distance round a circle is its circumference, which is 2 × π × r.
- The time for one orbit is the time period, T (the orbital period from Orbits of Planets, Moons, Comets and Satellites).
Put these into speed = distance ÷ time and you get the equation for orbital speed:
The equation
orbital speed = (2 × π × orbital radius) ÷ time period
v = (2 × π × r) ÷ T
v is in metres per second (m/s), r is in metres (m) and T is in seconds (s). This equation is given on the formulae page in the exam, so you do not need to memorise it, but you must be able to use it. Use the π button on your calculator.
Key terms:
- Orbital speed: how fast an object moves along its orbit, in m/s.
- Orbital radius: the distance from the centre of the object being orbited to the orbiting object.
- Circumference: the distance all the way round a circle.
Measuring the orbital radius
The orbital radius is always measured from the centre of the planet or the Sun, not from its surface. For the Earth going round the Sun, r is the distance from the Sun's centre to the Earth.
A satellite is usually described by its height above the ground. Then you must add the radius of the planet:
orbital radius = radius of the planet + height above its surface
Getting the units right
The equation only works in metres and seconds. Space questions give big numbers in other units, so convert first.
| Change | How |
|---|---|
| km to m | multiply by 1000 (6400 km = 6 400 000 m = 6.4 × 106 m) |
| minutes to s | multiply by 60 |
| hours to s | multiply by 3600 |
| days to s | multiply by 24 × 3600 = 86 400 |
| 1 year | 365 days = 365 × 86 400 = 3.15 × 107 s |
Standard form keeps big numbers tidy. Write 150 million km as 150 000 000 km = 1.5 × 108 km = 1.5 × 1011 m.
Worked examples
Worked example 1: a satellite
A satellite orbits 600 km above the Earth's surface. The radius of the Earth is 6400 km. One orbit takes 100 minutes. Calculate its orbital speed.
Radius: r = 6400 + 600 = 7000 km = 7.0 × 106 m
Time: T = 100 × 60 = 6000 s
v = (2 × π × r) ÷ T
v = (2 × π × 7.0 × 106) ÷ 6000
v = 7300 m/s (to 2 significant figures)
Worked example 2: the Moon
The Moon orbits the Earth at a radius of 384 000 km. One orbit takes 27 days. Calculate the Moon's orbital speed.
r = 384 000 km = 3.84 × 108 m
T = 27 × 86 400 = 2 332 800 s = 2.33 × 106 s
v = (2 × π × 3.84 × 108) ÷ (2.33 × 106)
v = 1000 m/s (to 2 significant figures)
Worked example 3: the Earth around the Sun
The Earth orbits the Sun at a radius of 1.5 × 1011 m. One orbit takes 365 days. Calculate the Earth's orbital speed.
T = 365 × 86 400 = 3.15 × 107 s
v = (2 × π × 1.5 × 1011) ÷ (3.15 × 107)
v = 30 000 m/s = 3.0 × 104 m/s (to 2 significant figures)
Rearranging for T and r
Sometimes the speed is given and you must find the time period or the radius. Rearrange first, then substitute.
| Find | Rearranged equation |
|---|---|
| Orbital speed | v = (2 × π × r) ÷ T |
| Time period | T = (2 × π × r) ÷ v |
| Orbital radius | r = (v × T) ÷ (2 × π) |
Worked example 4: finding T
A satellite has an orbital radius of 4.2 × 107 m and an orbital speed of 3100 m/s. Calculate its time period in seconds.
T = (2 × π × r) ÷ v
T = (2 × π × 4.2 × 107) ÷ 3100
T = 85 000 s (to 2 significant figures), which is about 24 hours.
Worked example 5: finding r
A space probe orbits a planet at 2000 m/s. One orbit takes 10 hours. Calculate the orbital radius.
T = 10 × 3600 = 36 000 s
r = (v × T) ÷ (2 × π)
r = (2000 × 36 000) ÷ (2 × π)
r = 1.1 × 107 m (to 2 significant figures)
Common mistakes
- Leaving the radius in km or the time in hours or days. Convert to m and s first.
- Forgetting the 2 in 2 × π × r. That gives only half the circumference.
- Measuring r from the surface of the Earth. Add the Earth's radius to the height so r runs from the centre.
- Pressing the calculator keys in the wrong order. Work out the top line, then divide by T. Put brackets round the bottom line if it has more than one number.
- Rearranging wrongly. For r you divide by 2 × π, not multiply.
Exam-style question
A weather satellite orbits the Earth at a height of 1600 km above the Earth's surface. The radius of the Earth is 6400 km. The satellite takes 2.0 hours for one orbit.
(a) Calculate the orbital radius of the satellite in metres. [2 marks]
(b) Calculate the orbital speed of the satellite. [3 marks]
(c) A second satellite orbits at a speed of 4000 m/s with an orbital radius of 2.5 × 107 m. Calculate its time period in seconds. [3 marks]
Model answer
(a) r = 6400 + 1600 = 8000 km (1); 8000 km = 8.0 × 106 m (1).
(b) T = 2.0 × 3600 = 7200 s (1); v = (2 × π × 8.0 × 106) ÷ 7200 (1); v = 7000 m/s (accept 6980 m/s) (1).
(c) T = (2 × π × r) ÷ v (1); T = (2 × π × 2.5 × 107) ÷ 4000 (1); T = 39 000 s (accept 3.9 × 104 s) (1).
Exam tip
Write each line in order: the equation, the numbers in metres and seconds, then the answer with its unit. Even if you slip on the calculator, you still collect the method marks.