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A Level Maths: Newton’s Laws
🧠 A Level Maths Newton’s Laws — A Teacher’s Walkthrough
When you reach Mechanics, you can’t go more than a couple of pages without meeting Sir Isaac Newton. His three laws appear across every exam board — Edexcel, AQA, OCR — and they sit underneath almost every topic: motion, slopes, pulleys, friction, projectiles. Once you see how the laws fit together, the whole subject begins to behave itself.
🔙 Previous topic:
“Return to the motion equations before introducing forces.”
🧠 Seeing the Bigger Picture
Students often memorise equations before they understand what they mean. The trick is to see each law as a story about motion. Newton wasn’t inventing rules for you to recite — he was describing how things behave when forces act. Once you grasp that, the maths just explains what you already imagine.
⚙️ Newton’s First Law — When Nothing Really Happens
A body stays as it is — either still or moving steadily — unless something unbalanced pushes or pulls it.
If the forces cancel out, the object keeps doing whatever it was doing. That’s equilibrium.
📏 Think of a book on a desk. Gravity pulls down, the desk pushes up equally, so the book rests there. If you slide it, it moves for a moment until friction adds that unbalanced force.
Whenever you read “moves with constant velocity,” that phrase screams equilibrium. Write a = 0 beside it — that’s easy credit.
✏️ Example 1 – Block on a Slope
A three-kilogram block slides down a slope with friction, coefficient 0.5, at steady speed.
“Steady” → no acceleration → forces balance.
Down-the-slope component of weight (3 g sin θ) equals friction (μR = 0.5 × 3 g cos θ).
That gives tan θ = 0.5, so θ ≈ 27 degrees.
✅ Exam tip: equilibrium always means the driving and resisting forces match.
❗ Common error: skipping the components and just writing 3 g = F.
🧠 Teacher Insight
Equilibrium isn’t a trick — it’s simply “no resultant.”
If all the arrows on your diagram cancel, you’re already using the first law.
⚙️ Newton’s Second Law — When Things Start Moving
Here’s the powerhouse: F = m a.
It says acceleration happens when there’s an unbalanced force, and the size of that acceleration depends on both the total push and the mass.
🧩 The word “total” is crucial. Always combine every force in that direction before using the formula.
✏️ Example 2 – Straightforward Start
An eight-kilogram object feels a horizontal force of ten newtons.
Ten = eight × a → a = 1.25 m/s².
Simple — but still write the reasoning; clarity earns marks.
✏️ Example 3 – Adding Resistance
A two-kilogram block is pulled by ten newtons while friction resists with four.
Resultant = ten − four = six.
Then six = two × a → a = 3 m/s².
❗ Common trap: putting a single force straight into F = m a without finding the resultant.
🧠 Teacher Aside
When I mark work, I can tell instantly who drew the diagram. Those students have consistent signs and fewer slips. A quick sketch at the start saves half the algebra pain.
✏️ Example 4 – Pulling at an Angle
A twenty-kilogram block is pulled across a rough floor by a string at thirty degrees.
Tension = fifty newtons, acceleration = 0.5 m/s².
Horizontally, fifty cos thirty − F = twenty × 0.5, so F ≈ 33 N.
Vertically, R + fifty sin thirty = twenty g, so R ≈ 168 N.
Hence μ = F / R ≈ 0.2.
🧠 Notice how the upward pull reduces R and therefore reduces friction.
❗ Don’t assume R = m g whenever there’s an angle.
✏️ Example 5 – Forces as Vectors
A five-kilogram particle experiences two forces: (4 i + j) and (− i + j).
Resultant = (3 i + 2 j).
Divide by five: a = (0.6 i + 0.4 j).
Magnitude ≈ 0.72 m/s².
📏 Keep it in i-j form until the end; direction often earns its own mark.
⚠️ Frequent Pitfalls
• Mixing up sine and cosine.
• Swapping signs mid-calculation.
• Forgetting that friction opposes potential motion.
• Assuming R = m g on a slope.
⚙️ Newton’s Third Law — The One About Pairs
“For every action, there is an equal and opposite reaction.”
Two forces, equal in size, opposite in direction, acting on different objects.
✏️ Example 6 – Box on a Table
A five-kilogram box sits still. Weight = 49 N down, reaction = 49 N up.
Press down with 10 N → R = 59 N.
Lift with 10 N → R = 39 N.
❗ Classic slip: drawing both forces on the same body.
Each acts on a different object — that’s the third law’s whole point.
🧠 Connecting the Three
First law → balance.
Second law → what happens when balance breaks.
Third law → how forces come in pairs.
Every mechanics problem uses all three somewhere: slopes, pulleys, friction, connected masses — same pattern, new setting.
📏 Exam-Day Routine
1️⃣ Draw the forces.
2️⃣ Pick a positive direction and stick to it.
3️⃣ Resolve angled forces parallel and perpendicular.
4️⃣ Write F = m a in each direction — even if a = 0.
5️⃣ Use the keywords: “smooth,” “light string,” “constant speed.”
Even when numbers go wrong, that method earns you marks.
❗ What Trips Students Up
- Mixing trig ratios.
- Forgetting that reaction is less than m g on a slope.
- Ignoring friction’s direction.
- Dropping units at the end.
Highlight those clues in the question — they exist for a reason.
🚀 Final Thoughts
If there’s one idea to hold onto, it’s this: Newton’s Laws aren’t a set of tricks to remember — they’re a way of seeing how the world moves. Once you grasp how forces and motion link together, problems stop feeling abstract. You’re just describing what actually happens when things push, pull, or speed up.
When you get a mechanics question, slow down. Sketch it out. Ask yourself, what’s really going on here? Once the story makes sense, the maths follows naturally.
And if you’d like a bit of structure while you revise, check out our Year 13 Maths Revision Course. It walks you through pure, stats, and mechanics in plain English, step by step. Perfect if you want Newton’s Laws — and the rest of A Level Maths — to finally click.
About the Author
S. Mahandru is the Head of Mathematics at Exam.tips, specialising in A Level and GCSE Mathematics education. With over a decade of classroom and online teaching experience, he has helped thousands of students achieve top results through clear explanations, practical examples, and applied learning strategies.
Updated: November 2025
🧭 Next topic:
“Now, learn how to resolve forces and apply Newton’s second law.”
