Do I Need To Take A-Level Maths To Do A-Level Physics?

People often say to me, "Dave, do I need to take A-level maths if I want to do A-level physics?"  Not generally random strangers, mind;  usually Year 11 students, or their parents.

Most schools will tell you emphatically YES.  Some even make it mandatory.  In my experience this is a lot of tosh.

Let's spell it out:  you do NOT need to take A-level maths in order to do well in A-level physics.  A-level physics is full of maths.  But although there's lots of it, there are actually very few mathematical concepts you need to master in order to study A-level physics.

Nearly all of them you will have learned at GCSE (possibly not all that well, but you'll have covered them).  It's really important that you can do these few things well - and you understand how they work, and why.  The rest is just wallpaper.

In fact I'd contend that you don't even need to have studied GCSE maths to understand A-level physics - you could simply learn this handful of things at the beginning of your course, if you had to, although I'd not generally recommend this, as you need to be adept at using them.

Key Skill 1:  re-arranging the subject of an equation

This comes up all the time.  A good GCSE maths teacher will have taught you not only how to do it, but why you do what you do.  I've always been amazed at how few Y12 students really understand how this works;  it's surprisingly simple.  All you have to do is remember three simple rules of equations, which I deal with below.

If you can re-arrange the subject of the following equations for each letter in term, you'll be able to cope with every equation in A-level physics bar one tiny set (see below).
e = ½ m v²
v² = u² + 2as

Key Skill 2:  exponential numbers

This comes up all the time in physics, because we often deal with very large or very small numbers.  For example, the charge of an electron is 1.6 x10^-19 coulombs.  We could write 0.00000000000000000016 coulombs, but we're lazy scientists, so we don't.

You need to understand how the exponential annotation works, but critically, you also need to understand how to use it on your damn calculator!  So many times I've seen highly competent physics students get the wrong answer because they don't know how to use their calculator.  Clue:  use the "EE" or "Exp" button - do NOT type in [1.6] [x] [10] [^] [-19] for example, because if you're in the middle of an equation, your calculator won't know which order to do things in, and you are guaranteed to get the wrong answer.

To see what I mean, calculate z:
z = 3x10^8 / 1x10^3
(Excuse me using ^ instead of superscript - I can't see how to format superscript text here...)
If you get 300,000 (or 3x10^5), you got it right.  If you get 3x10^11, then give yourself a smack on the back of the head, and go and learn to use your calculator properly.

Key Skill 3:  trigonometry

Make sure you can use sine, cosine and tangent;  they come up in most things involving more than one dimension, and when you do simple harmonic motion, you'll begin to understand some of the beauty of them - including some elegance to do with differentiation and integration.

The rest

At A2 you'll need to do a couple of logarithms. Quite honestly, you don't really need to understand logs all that well to get the answers right, although it would be nice - you can just learn the basic rules of how to rearrange these equations. This is primarily when doing half-life and capacitors.

If you are skilled at mental arithmetic, you'll find physics much more straightforward and rewarding. Consider this:

• 8x10^12 / 2x10^8

If you needed a calculator to work this out, consider that this is the same as writing 8/2 x 10^12/10^8.  Well, 8 / 2 = 4, and 10^12 / 10^8 = 10^4, so the answer is simply 4x10^4.

On this note, the ability to quickly estimate (which I learned so well studying Earth Sciences at university) is valuable and will help you avoid making silly mistakes. For example, if a circle has diameter 1.5m, its circumference is about 5 metres, so if something moves round this circle at roughly one rotation per second, its speed is roughly 5m/s. For the purposes of quickly working out the speed of a bunsen burner swinging around on the end of a hose, I don't really care whether pi is 3.1415926535... or 3.14 or 3. 3 and a bit will do. So pi times the diameter is just over 3 x 1.5m, which in my head is not far short of 5m. In an exam I'd work it out; but if I can quickly estimate, I can check my answer mentally and make sure I haven't done something silly with a calculator.

And for heaven's sake make sure you know how to plot a scatter graph (line graph, x-y graph). Clue: the numbers on each axis should go up the same amount each time. You'd be amazed how many A-level physics students forget this at some point.

All you really need to know about equations

So let's go back to these equations.  There are only three things you really need to understand about equations.

Rule 1:  each letter just stands for a number.  Sometimes we know what the number is.  Sometimes we want to find out.  Sometimes the number can only be expressed in terms of other numbers.  But any letter in an equation just stands for a number.

• y = 3 + 2

This is pretty obvious.  We can work out the value of y.  But have you also considered the following:

• distance = 3 km

3 is a number.  k stands for a number:  specifically a thousand.  m also stands for a number;  but one that we can only really express in terms of other numbers, such as speed and time.  So we normally leave m as m.  But we can substitute 1000 for the value of k:

• distance = 3 km = 3 x 1000 x m = 3000m

It sounds obvious when you think about it.  But how many people understood this when they did GCSE maths?
When you have to deal with units of MeV/c², if you know that M means 10^6, one eV = 1.6x10^-19 J, and c² is 9x10^16, you'll breeze it.

Rule 2:  providing you do the same thing to both sides of an equation, it's still valid.  If you do different things to each side, you'll break it.

Rule 3:  anything divided by itself is one.  Anything times one is itself.  This is really useful when we want to re-arrange the subject of an equation.

Example:  we have the formula I = V / R (current = voltage / resistance).  Let's say we need to re-arrange this for V.

It doesn't matter if you can do this straight away in your head:  what's important is what you do and why.

If at this point you thought "ah, we're dividing by R on the right hand side, so we'll times by R on the left hand side", then find whoever taught you how to re-arrange equations and deliver a short, sharp blow to their temple.

We want to get rid of R on the right hand side of the equation.  We know (Rule 3) that anything divided by itself is 1.  So let's multiply the right hand side by R, so we can later divide it by itself to get 1, and get rid of it.

If we're going to do that, we need to do the same to the left hand side (Rule 2).

So we get:

• I R = V R / R

R / R is 1 (Rule 3) and V x 1 is V (Rule 3).  So V R / R is the same as V.

Now we have:

• I R = V

The equals sign works both ways, so if I R = V then V = I R.

Not hard, is it?

We don't often add or subtract things in physics (multiplication and division are seemingly more relevant to the laws of the universe), but if we have equations which have addition or subtraction, simply remember that anything minus itself is zero, anything plus zero is itself, and deal with the addition / subtraction before you divide or multiply.

If you followed the last few paragraphs, then take it from me, you can re-arrange any equation that is going to come up in A-level physics, apart from the logarithm stuff mentioned above, which you can learn as you go along.