## 3. Compound units

If you have already looked at earlier pages in this document, you will have seen seen that some of the quantities that you will meet do not have simple units. For instance, speed is distance divided by time - metres per second or kilometres per hour. To represent these measurements, we need to use compound units.

In considering length, we mentioned that area is measured in square metres and we wrote this as m^{2}. This is shorthand for m × m (metres times metres) - the sum that you would do to work out an area. We can use the same notation to show speed. Here our distance unit is metres and our time unit is seconds. Our sum is m ÷ s, which is written as m s^{-1} (metres per second), not as m/s. Note that the separate units are divided by a space: '1.75 m s^{-1}' indicates a speed of 1.75 metres per second, whilst '1.75 ms^{-1}' represents a frequency of 1.75 events per millisecond (ms).

We may be more interested in acceleration than speed. Acceleration is the rate at which speed changes with time: m ÷ s ÷ s. The unit for acceleration is m s^{-2} (metres per second per second). Some common quantities and example units are given in the following table:

### Table of common compound quantities and example units

Measurement | Calculation | Example unit |

Area | length × length | m^{2} |

Volume | area × length | m^{3} |

Density | mass ÷ volume | kg m^{-3} |

Speed | length ÷ time | m s^{-1} |

Acceleration | speed ÷ time | m s^{-2} |

Energy transfer rate | energy ÷ time | J s^{-1} = W |

Energy transfer rate per unit area | energy ÷ time ÷ area | J s^{-1} m^{-2} = W m^{-2} |

Force | mass × accleration | kg m s^{-2} = 1 N |