Quantities, Units and Reading a Physics Question

Almost every physics item you will meet is built from a handful of quantities and their units, and a surprising number of wrong answers are not physics errors at all — they are unit errors, or a decimal point in the wrong place. Before anything else, then, it is worth being able to say what a quantity is, what its unit is, and whether it carries a direction.

Quantities and their units

A quantity is something you can measure — a length, a mass, a time — and a unit is the agreed amount you measure it against. The system used throughout these tests is SI, which builds everything from a small set of base units: the metre for length, the kilogram for mass, the second for time. Everything else is assembled from those. Speed is a length divided by a time, so its unit is metres per second. Nothing about that is arbitrary, and it is the reason you can often reconstruct a unit you have forgotten rather than having to recall it.

Prefixes, and what they actually do

Units are often too big or too small to be convenient, so SI attaches a prefix that multiplies or divides them by a power of ten. Kilo- means a thousand times, so a kilometre is a thousand metres and a kilogram is a thousand grams. Centi- means a hundredth, so a centimetre is a metre divided by a hundred. Milli- means a thousandth, so a millimetre is a metre divided by a thousand, and a millisecond is a thousandth of a second. Mega- means a million. These are the four you will actually need. The point to hold onto is that a prefix is not part of the quantity — it is only a statement about the size of the unit you chose to measure it in, which is why the same length is 2 metres, 200 centimetres and 2000 millimetres without anything about the length itself having changed.

Converting between them is a matter of deciding whether the number should get bigger or smaller, and it is worth doing that decision consciously rather than by reflex. Going from metres to centimetres, the unit gets smaller, so it takes more of them to cover the same distance and the number must get bigger: multiply by 100. Going from centimetres to metres, the unit gets bigger, so you need fewer and the number must get smaller: divide by 100. If you check the direction that way every time, you will not have to remember which conversions multiply and which divide, because you will be deriving it.

Building a unit from the definition

Take force. Newton's second law says force is mass times acceleration, and acceleration is a change of speed divided by a time — metres per second, per second. So force must be measured in kilogram metres per second squared. That combination is given its own name, the newton, purely for convenience: one newton is the force that accelerates one kilogram at one metre per second squared. If an item offers you the joule, the watt and the pascal alongside the newton and asks which measures force, you do not need to have memorised the list — you need only rebuild it from F = ma.

The phrase "per second squared" deserves unpacking, because it looks stranger than it is. Speed is metres per second. Acceleration is how fast the speed is changing, so it is metres per second, per second — a change in speed, divided again by the time it took. Writing that twice-divided second as "second squared" is only shorthand for having divided by time twice. Nothing is being multiplied by itself in any physical sense.

That summary is worth pausing on, because the same three quantities will come back in three different arrangements, and the next section is about moving between them.

Rearranging a formula

Nearly every calculation in this course comes down to taking a relationship and solving it for whichever quantity the question left out, so it is worth being explicit about the mechanics rather than assuming them. The single rule is that whatever you do to one side of an equals sign, you must do to the other, and the aim is to get the quantity you want alone on one side. Start from $\text{speed} = \frac{\text{distance}}{\text{time}}$. Suppose you know the speed and the time and want the distance. Multiply both sides by time: on the left you get speed × time, and on the right the time in the numerator and the time in the denominator cancel, leaving distance. So distance = speed × time.

Now suppose instead that you know the speed and the distance and want the time. Start from the version you just derived, distance = speed × time, and divide both sides by speed. On the right, the speeds cancel and leave time; on the left you have $\frac{\text{distance}}{\text{speed}}$. So $\text{time} = \frac{\text{distance}}{\text{speed}}$. Those three arrangements are the same single fact written three ways, and the same procedure works on every formula in this course — F = ma becomes $m = \frac{F}{a}$ and $a = \frac{F}{m}$ by exactly the steps above. This is why it is worth learning the move rather than memorising each rearrangement separately: three formulas become one, and the triangle mnemonics become unnecessary.

Direction: vectors and scalars

Some quantities are fully described by a number and a unit, and some are not. Mass, time and temperature are complete as they stand: two kilograms, thirty seconds, twenty degrees. These are scalars. Others are meaningless until you say which way they point — a force of ten newtons pushing left does something quite different from ten newtons pushing right. These are vectors, and the ones you will meet most are force, velocity, acceleration and momentum. The pairing that catches people out is speed and velocity: they are measured in the same unit and often have the same number, but velocity is speed with a direction attached, and speed is velocity with the direction thrown away. A car going round a roundabout at a steady thirty kilometres per hour has a constant speed and a continuously changing velocity.

Putting it to work

The three examples below use the same single relationship in all three of its arrangements, which is the point: there is one fact here, not three to memorise.

That last step is worth dwelling on, because it is the cheapest error-catch available to you. Had you divided by mistake in Example 2, you would have got $\frac{40}{30}$, about 1.3, and the units would have come out as metres per second per second — an acceleration, plainly not a distance. Whenever an answer's units do not match the quantity the question asked for, an operation went the wrong way round, and you can know that without checking the arithmetic at all.

Decimal discipline

The arithmetic in these tests is deliberately light — clean numbers, no calculator — which means the errors that survive are structural rather than computational. Dividing when you should multiply gives an answer that is wrong by a factor of the square of whatever you divided by, and it will usually be sitting there among the options waiting for you. So will the same digits with the decimal point moved. When an answer looks implausible for the situation described — a walking pace of 150 metres per second, a car mass of 12 kilograms — that feeling is worth trusting; it almost always means an operation went the wrong way round.

It helps to carry a few reference values so that implausible answers announce themselves. A brisk walk is about 1.5 metres per second. A car on a motorway is about 30 metres per second. Sound travels at about 340 metres per second and light at about 300 million. A bag of sugar is a kilogram, an adult is somewhere near 70, a small car is around 1000. None of these are things you will be asked for directly; they are the yardsticks that let you notice when an answer is out by a factor of a thousand, which is a far more common failure than being out by a few per cent.

Why it matters in aviation

Aviation is one of the few technical fields that deliberately keeps a mixed unit set: altitude in feet, speed in knots, distance in nautical miles, atmospheric pressure in hectopascals or inches of mercury, and fuel in either kilograms or pounds depending on where the aircraft was built and where it is operated. Each of those units survived because it suited the people who needed it, and the navigational ones are not arbitrary at all — a nautical mile is one minute of latitude, so a degree of latitude is sixty of them and the latitude scale on a chart doubles as a distance scale.

The consequence is that unit discipline in flying is a safety matter rather than a point of pedantry. A fuel quantity loaded in pounds when the flight plan assumed kilograms is well under half the intended mass, and the aircraft leaves with far less endurance than the crew believe they have. The habit this lesson asks for — naming the unit alongside every number, and deciding consciously whether a conversion should make the figure larger or smaller — is the same habit that catches that error before the aircraft moves.

What the test asks

Expect items that give you a definition and ask for the unit, or give you a unit and ask which quantity it measures, and expect at least one item asking you to sort a list into vectors and scalars. The reliable trap is offering speed as a vector or velocity as a scalar, so fix the distinction now: if removing the direction would change nothing about the answer, it is a scalar.