Welcome to Mathematical Skills in Physics!

Physics is often called the "study of everything," from the smallest atoms to the entire universe. While that sounds huge, we use a very specific language to describe how these things work: Mathematics. Don't worry if you aren't a "maths person" yet—this guide is designed to break down the tools you need for your AQA Combined Science: Trilogy exams into simple, manageable steps.

Note: This chapter focuses on the skills you need across all your Physics papers. For specific details on how to set up experiments, check out the Required Practicals chapters.

1. Working with Numbers

In Physics, we deal with things that are incredibly tiny (like the size of an atom) and things that are massive (like the energy from the Sun). To handle these, we use a few numerical tricks.

Standard Form

Standard form is a way of writing very large or very small numbers so they are easier to read. It always looks like this: \(A \times 10^n\).

For example, the speed of light is \(300,000,000 \text{ m/s}\). In standard form, we write this as \(3.0 \times 10^8 \text{ m/s}\).
If the power (\(n\)) is positive, the number is big. If it is negative (e.g., \(10^{-3}\)), the number is small (less than 1).

Significant Figures (sf)

When you calculate an answer, your calculator might give you a long string of decimals like \(4.5672384\). In Physics, we usually round this to 2 or 3 significant figures.
Top Tip: Always look at the numbers given in the question. If the question uses 2 significant figures, give your answer to 2 significant figures!

Quick Review: Symbols You Must Know

\(=\) : Equal to
\(\approx\) : Approximately equal to
\(\propto\) : Proportional to (as one goes up, the other goes up)
\(<\) : Less than
\(<<\) : Much less than
\(>\) : Greater than
\(>>\) : Much greater than

Key Takeaway: Standard form and significant figures make messy numbers clean and professional. Always check if the question asks for a specific number of decimal places or significant figures!

2. Mastering Equations

For your 2027 and 2028 exams, AQA will provide an equation sheet with all the physics formulas. You don't need to memorise them, but you must know how to choose the right one and use it correctly.

The FIFA Method

This is a foolproof way to solve any calculation question:

  1. F (Formula): Find the equation on your sheet that has the quantities mentioned in the question. Write it down.
  2. I (Insert): Put the numbers from the question into the formula (substitution).
  3. F (Fine-tune): Rearrange the equation if the thing you are looking for isn't by itself.
  4. A (Answer): Calculate the final result and don't forget the Units!

Rearranging Equations

Sometimes the equation isn't in the format you need. Think of an equation like a pair of balanced scales. Whatever you do to one side, you must do to the other.

Example: To find mass from \(F = m \times a\):
We want \(m\) on its own. Since it is multiplied by \(a\), we do the opposite and divide both sides by \(a\).
\(m = \frac{F}{a}\)

Units are Vital!

In Physics, a number without a unit is just a squiggle. You must know the standard SI units:

  • Mass: kilograms (\(\text{kg}\))
  • Distance: metres (\(\text{m}\))
  • Time: seconds (\(\text{s}\))
  • Force: Newtons (\(\text{N}\))
  • Energy: Joules (\(\text{J}\))
  • Power: Watts (\(\text{W}\))

Common Mistake: Forgetting to convert units! If the mass is in grams (\(\text{g}\)), divide by 1000 to get kilograms (\(\text{kg}\)) before you start your calculation.

3. Graph Skills

Graphs tell a story of how two things are related. You will often be asked to plot data or interpret a graph.

The Gradient (Slope)

The gradient tells you the rate of change. For a straight-line graph:
\(y = mx + c\)
Where \(m\) is the gradient and \(c\) is the y-intercept (where the line crosses the vertical axis).

To find the gradient (\(m\)):
\(m = \frac{\text{change in } y}{\text{change in } x}\)

Tangents to Curves

If a graph is curved, the gradient is constantly changing. To find the rate of change at a specific point, you must draw a tangent.
How to do it: Draw a straight line that just touches the curve at that exact point. Then, calculate the gradient of that straight line.

Area Under the Curve

In some Physics graphs (like Velocity-Time graphs), the area under the line represents a value (like distance).
Did you know? If the shape under the line is irregular, you can estimate the area by counting the squares on the graph paper.

Key Takeaway: Use a sharp pencil for graphs. A straight line of best fit should have an equal number of points above and below the line.

4. Geometry and Shapes

Physics often involves 2D and 3D shapes, especially when calculating density or pressure.

  • Area of a rectangle: \(\text{width} \times \text{height}\)
  • Area of a triangle: \(0.5 \times \text{base} \times \text{height}\)
  • Volume of a cube/column: \(\text{length} \times \text{width} \times \text{height}\)

Higher Tier Tip: You may need to handle vector quantities. Remember that scalars only have size (like speed), but vectors have size AND direction (like velocity).

5. Summary Checklist

Before you head into your exam or practical, make sure you can:

  • Convert numbers into standard form.
  • Round answers to the correct number of significant figures.
  • Calculate the mean from a set of data (add them all up and divide by how many there are).
  • Substitute numbers into a formula from the equation sheet.
  • Change the subject of an equation (rearrange it).
  • Draw a line of best fit on a scatter graph.
  • Calculate a gradient from a straight line.

Don't worry if this seems tricky at first! The more you practice using the equations from your provided sheet, the more natural it will feel. Physics is just about using these tools to solve puzzles!