Average orbital speed: one lap round, divided by the time it took

1. Speed is always distance divided by time, so the only real work in this topic is writing down the distance an orbiting object covers.

2. An orbit is treated as a circle of radius $r$, measured from the centre of the orbit out to the orbiting object.

3. The distance covered in one complete orbit is therefore the circumference of that circle.

   $$distance\; in\; one\; orbit=2\pi r$$

4. The time for that one orbit is the orbital period $T$, so dividing the circumference by $T$ gives the average speed.

   $$v=\frac{2\pi r}{T}$$

5. The word average matters, because a real orbit is slightly squashed and the object speeds up and slows down as it goes round; this equation gives the speed averaged over the whole lap.

6. You must be able to write the equation down from memory, since no equation list is handed out with the paper.

Units first: metres and seconds, or the answer is wrong by thousands

1. For a speed in $m/s$, put $r$ in metres and $T$ in seconds before substituting anything.

2. Orbital periods are almost always quoted in days, hours or years, so a conversion is nearly always the first step.

   $$1hour=3600s$$ $$1day=24\times 3600=86400s$$

3. Radii in space physics are enormous, so work in standard form throughout rather than typing long strings of zeros into the calculator.

4. The equation rearranges in the usual way if the question gives you the speed and asks for the period or the radius.

   $$T=\frac{2\pi r}{v}r=\frac{vT}{2\pi }$$

Reading the equation: what makes an orbiting object fast or slow

1. For the same orbital radius, a shorter period means a higher speed, because the same distance is covered in less time.
2. For the same period, a larger radius means a higher speed, because a bigger circle has to be covered in the same time.
3. In a real planetary system both quantities change together, so you cannot judge which planet is fastest by looking at radius alone: put the numbers into the equation and compare.
4. The equation works for anything that goes round anything else, so the same three symbols handle a planet round the Sun, the Moon round the Earth or a satellite round the Earth.