Key ideas: units, vectors and equations – GCSE Combined Science Physics

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Imagine a recipe that says add 2, and you have to ask: two what? Two spoons of sugar is lovely, but two kilograms would ruin the cake. The number means nothing until you know the unit.

Physics is the same. A distance of 6 could be 6 metres or 6 kilometres, and one is a short walk while the other is a long drive. Units are the words that give numbers their meaning.

Or think about directions. If a friend says walk 100 steps, you would ask which way, because 100 steps the wrong way gets you lost. Some quantities in Physics need a direction as well as a size, and you will meet those too.

This lesson is about the tools you use in almost every Physics question. They are units and prefixes, converting between units, standard form, scalars and vectors, rearranging equations and rounding sensibly.

None of it is hard, but small slips cost a lot of marks. Using grams instead of kilograms, or leaving the unit off, are the commonest ways to lose marks that you had really earned.

Think of this lesson as your toolbox, with a tool for each job. You will keep reaching into it in every other Physics topic, from forces to electricity to energy. Ten minutes spent getting these tools right now will save you hours later.

You do not need to be quick at maths to do well here. You need to be careful, to write each line down, and to look at the units every time.

The chart shows the routine that almost every calculation follows. If you follow it step by step, you will find calculations calmer, and you will make fewer silly mistakes.

By the end you will be able to do three things. You will convert units and write numbers in standard form. You will tell scalars from vectors, and you will rearrange an equation and give a sensible answer with its unit.

Units, symbols and prefixes

Every quantity in Physics has a symbol and a unit. Force is F, measured in newtons, N, and energy is E, measured in joules, J. Power is P, measured in watts, W, so a 2000 W kettle transfers 2000 J of energy every second.

Time is t, in seconds, s. Current is I, in amperes, A, which is often called amps, and potential difference is V, in volts, V. Potential difference is the voltage across a component, and a mains socket in the UK gives about 230 V.

Sometimes a unit is too small or too big for the job. A prefix is a short word in front of a unit that changes its size, and kilo, k, means 1000. So a 2 kW heater is a 2000 W heater, and a 5 km walk is 5000 m.

Milli, m, means one thousandth, so 5 mA is 0.005 A, which is a tiny current. Mega, M, means one million, and giga, G, means one thousand million. Centi, c, means one hundredth, micro, μ, means one millionth, and nano, n, means one thousand millionth.

Notice that the letter m can mean two things. On its own, m is the metre, and in front of a unit it means milli, so mm is millimetres and ms is milliseconds. Writing the unit neatly with every number is a habit that picks up easy marks.

Converting units

Equations only work when the units fit together. If a speed is in metres per second, the distance must be in metres and the time must be in seconds. So convert to base units (metres, kilograms, seconds and amps) before you substitute, and if you only remember one rule from this lesson, make it this one.

To change grams to kilograms, divide by 1000. To change centimetres to metres, divide by 100, and to change millimetres to metres, divide by 1000. Milliamps to amps is divide by 1000 as well, because milli means one thousandth, and these divisions come up again and again.

Going the other way, multiply. To change kilometres to metres, multiply by 1000. To change minutes to seconds, multiply by 60, and to change hours to seconds, multiply by 3600 because there are 60 × 60 seconds in an hour.

Here is a trick to check the direction. If you change to a smaller unit, you need more of them, so the number gets bigger, and a gram is smaller than a kilogram. If you change to a bigger unit, you need fewer of them, so the number gets smaller.

A very common slip is to use 25 cm in an equation that needs metres. Always stop and ask whether the units match before you press any buttons, because the calculator will happily give you a wrong answer without a warning.

Standard form

Some numbers in Physics are huge and some are tiny. The speed of light is 300 000 000 m/s, and a small mass might be 0.0004 kg. Writing all those zeros is slow, and it is easy to lose count of them.

Standard form is a tidy way to write them. It is a number between 1 and 10 multiplied by a power of 10, and the power of 10 tells you how far the decimal point has moved. So the speed of light is 3.0 × 10⁸ m/s, and 0.0004 kg is 4 × 10⁻⁴ kg.

To write a number in standard form, move the decimal point until you have a number between 1 and 10. Count how many places you moved it, because that count is the power of 10, and the sign of the power says which way you moved. Then decide whether the power is positive or negative: moving the point to the left gives a positive power, and to the right gives a negative one.

A big number gets a positive power, and a small number gets a negative power. For 6 000 000 you move the point 6 places, so it is 6 × 10⁶. For 0.0035 you move it 3 places the other way, so it is 3.5 × 10⁻³.

The first number must always be between 1 and 10. So 0.4 × 10⁻³ is not in standard form, and it should be written 4 × 10⁻⁴. A scientific calculator has an EXP or ×10ˣ button for typing these numbers in, so use it instead of typing the zeros.

Scalars and vectors

Some quantities only need a size. If I tell you the temperature is 20 °C, there is no direction to give, because a temperature does not point anywhere. A quantity with a size only is called a scalar, and the name comes from the idea of a scale.

Scalars include distance, speed, mass, time, energy and temperature. You can describe each one completely with a number and a unit, such as 70 kg or 12 s. Nobody needs to ask which way a mass points.

Other quantities need a direction as well. If I push you with 30 N, it matters a lot whether I push you forwards or backwards. A quantity with a size and a direction is called a vector.

Vectors include displacement, velocity, force, weight, acceleration and momentum. A vector can be drawn as an arrow, where the length of the arrow shows the size and the arrowhead shows the direction.

In the picture the trolley is pushed to the right with a big force and held back to the left with a smaller force. The longer arrow is the bigger force, and the two arrows do not point the same way, so they pull against each other. The difference between them is the resultant force, and it decides how the trolley moves.

Distance and displacement

Distance is how far you travel along the path you took, and it is a scalar. Displacement is the straight line from the start to the finish, together with its direction, so it is a vector. Think of the distance as what your step counter shows, and the displacement as how far you are from home and in which direction.

Imagine you walk 3 m east and then 4 m north. The distance you walked is 3 + 4 = 7 m. But the straight line from where you started to where you ended is shorter, 5 m, pointing partly east and partly north.

Speed and velocity work the same way. Speed is distance divided by time, so it is a scalar. Velocity is displacement divided by time, so it is a vector with a direction as well as a size.

This difference matters when you finish where you began. If you walk 3 m east and then 3 m back west, the distance is 6 m but the displacement is 0 m. Your average velocity for the whole trip is zero, even though you were moving the whole time.

When a question asks for displacement or velocity, always say which way. A good answer reads 5 m to the right or 4 m/s to the north, not just the number, because the direction is part of the answer.

Rearranging and rounding

An equation is like a balanced see-saw. Whatever you do to one side, you must do to the other, or it tips over. Do the same to both sides and it stays balanced.

Start with speed = distance ÷ time. Multiply both sides by time to get distance = speed × time, or divide both sides by speed to get time = distance ÷ speed. Each form is the same equation arranged differently, and you choose the one that has the unknown on its own, such as the time when you know the distance and the speed.

The same trick works for V = I × R. Divide both sides by R to get I = V ÷ R, or divide both sides by I to get R = V ÷ I. The triangle helps: cover up the quantity you want, and what is left shows how to work it out.

A good order for any calculation is to write the equation, rearrange it, substitute the numbers with their units, work it out and give the unit. Writing each line makes it easy for an examiner to give you marks, even if the final answer slips. A neat layout also helps you spot your own mistakes, and it makes it easier to check your work at the end.

Significant figures are the digits that matter, counted from the first digit that is not zero. So 0.04567 to 2 significant figures is 0.046. Give 2 or 3 significant figures unless told otherwise, and check the answer is sensible, because a person walking at 300 m/s means you should check your working.

Recap

Every answer needs a number and a unit. Force is in newtons, energy in joules and power in watts. Prefixes change the size of a unit, so kilo is 1000 and milli is one thousandth.

Convert to base units before you calculate. Divide to go from g to kg, cm to m and mA to A. Multiply to go from km to m, minutes to seconds and hours to seconds.

Standard form is a number between 1 and 10 times a power of 10. Big numbers have a positive power and small numbers a negative power.

Scalars have a size only, and vectors have a size and a direction. Distance and speed are scalars. Displacement and velocity are vectors, and a round trip can have a displacement of zero.

To rearrange an equation, do the same thing to both sides. Then substitute, work out the answer, round sensibly and give the unit.

Here is a worked mini-example to try in your head. A trolley moves at 2 m/s for 3 minutes. Change the time to 3 × 60 = 180 s, then distance = speed × time = 2 × 180 = 360 m.

Before you move on, test yourself with a quick one. Change 450 g to kilograms, write 25 000 in standard form, and say whether 4 m/s to the north is a scalar or a vector. The answers are 0.45 kg, 2.5 × 10⁴ and a vector.

Write a short checklist on your revision notes and tick it off for every calculation. Does each quantity have a unit, are all the units base units, and is the equation the right way round? Do the numbers give an answer of a sensible size, and did you write the unit at the end?

A last check you can always use is to ask whether the answer is sensible. Walking is about 1.5 m/s, cycling is about 6 m/s, and a car on a road is about 25 m/s. If your answer is far from these, look for a unit slip.

How this is asked in an exam: a question often hides a unit trap, such as minutes or grams, that you must convert before using the equation. A distance or displacement question wants you to say which one you are giving. A calculation wants the equation, the substitution and the answer with its unit.