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Scales in space and time
Scales in space and time

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 Compound units and rates

Some quantities need multiple units (known as compound units) to describe them. One such quantity is speed, which is a measure of distance travelled over time taken.

For example, at the time of writing (2017), the women’s world record for the 100.0 m sprint is 10.49 s, as set by Florence Griffith-Joyner in 1988. To calculate the (average) speed at which she ran, you need to divide the distance over the time:

multiline equation line 1 equation left hand side 100 m divided by 10.49 s equals right hand side 9.532 times m divided by s line 2 equals 9.532 m s super negative one

Her average speed was therefore 9.532 metres per second (m s−1). This value describes the average rate at which she covered distance over the course of the race, namely 9.532 metres every second.

Note the way the units are written in the example above. The result of the calculation could be reported as ‘metres per second’, ‘m per s’ or ‘m/s’, but ‘m s−1’ is most appropriate for scientific reporting as it is the most concise.

The notation of ‘negative exponents’ is commonly used for units. So, for example, just as equation sequence one divided by 25 equals one divided by five squared equals five super negative two, can be expressed one divided by s as s−1 and one divided by m super three as m−3. Writing compound units using negative exponents is generally good scientific practice.

You should also note that care is needed when writing and interpreting compound units. A space is required between the ‘m’ and the ‘s’ in m s−1. The unit ms−1 has a completely different meaning, namely ‘per millisecond’.

Any quantity that changes with time can be expressed as a rate. For example, it takes 5 minutes for an oven to heat up from room temperature, 20 °C, to 200 °C. The change in temperature is 180 °C, so the (average) rate at which the temperature changes is:

multiline equation line 1 equation left hand side 200 postfix degree cap c negative 20 postfix degree cap c divided by five min equals right hand side 180 postfix degree cap c divided by 300 s line 2 equation left hand side equals right hand side 0.6 postfix degree cap c s super negative one

Usually SI units should be used for calculations, which is why minutes were converted to seconds in the example above. However, at times, other non-SI units might be more meaningful. For example, reporting the oven temperature in degrees Celsius (°C) is far more familiar than using the SI unit of temperature – kelvin, K.

It is important to include units throughout a calculation, as well as in a final answer. It makes it clear to the reader (and yourself!) what units are used, and where units have been converted.

Don’t forget, if you need any guidance on the maths content, take a look at the badged open course, Mathematics for science and technology [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .