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Variable Acceleration in A Level Mechanics
Variable Acceleration in A Level Mechanics — Why It Looks Scary but Really Isn’t
Every September someone says, “Sir, I get SUVAT… until it stops working.”
That, right there, is the doorway into variable acceleration.
When acceleration isn’t constant, you don’t throw the rules away—you just zoom out and look at the motion through calculus glasses.
It’s the same story, just with smoother transitions.
🔙 Previous topic:
“Review 10 essential mechanics tips before tackling variable acceleration.”
🔁 The calculus ladder (your new SUVAT)
📏 Displacement s(t)
📏 Velocity v(t) = ds/dt
📏 Acceleration a(t) = dv/dt = d²s/dt²
So: differentiate going up the ladder, integrate coming down.
That’s it.
In class I usually draw the ladder, mutter something like “Up means sharper, down means gentler,” and suddenly half the room nods.
🧩 Why it matters
Constant acceleration is the tidy textbook world.
Variable acceleration is real life—cars speeding up unevenly, skydivers feeling drag, planets doing their elliptical thing.
And the exam boards—AQA, Edexcel, OCR—love asking about it because it shows who can actually link algebra to motion.
🔁 The four-beat method I swear by
1️⃣ Decode what you’re given. Is it a(t)? v(t)? Maybe just a v–t graph?
2️⃣ Choose your move.
a → ∫ to v → ∫ to s
v → differentiate for a, integrate for s
Graph → gradients go up the ladder, areas go down
3️⃣ Use initial clues. “Starts from rest” means v(0)=0; “through origin” means s(0)=0.
4️⃣ Answer the question not the algebra. Find where v=0 (turning points), a=0 (max speed), and split distances when direction changes.
✅ OCR phrase that wins a mark: “Since acceleration isn’t constant, integrate a(t) to find v(t) using the given conditions.”
📏 Real-world sense check
Think about pressing a car pedal. Acceleration starts big, then fades as drag fights back. That’s variable a.
Differentiate to find the “feel” of speed, integrate to see how far you’ve gone.
🧠 Funny moment: last year one student graphed v vs t, shaded the area, and said, “Wait — that’s literally distance!” That spark = understanding.
❗ Board quirks worth remembering
- AQA: hides data like “passes through origin” in the opening line—use it before you integrate.
- Edexcel: throws in “total distance” to see who splits intervals at v=0.
- OCR: goes graphical — say “slope and area,” and you pocket a method mark.
🧠 Tiny teacher confession
I once marked six perfect integrations—all missing + C. They lost two marks each.
Now my classes write “+ constant (find it later)” in bright pink. It’s become a meme. Still, nobody forgets it now.
📚 Example 1 – Edexcel-style distance question
a(t)=4t−8, v(0)=6, s(0)=0
Velocity → v = ∫(4t−8) dt = 2t²−8t + C, so C=6.
✅ v(t)=2t²−8t+6.
Displacement → s = ∫v dt = (2/3)t³−4t²+6t + K, K=0.
Set v=0 → 2t²−8t+6=0 → (t−1)(t−3)=0 → t=1, 3 s.
s(1)=8/3 m, s(3)=0 m.
Total distance = |0−8/3| = 2.67 m.
✅ Write the sentence “Total distance = sum of absolute displacements” — Edexcel’s own wording.
📚 Example 2 – OCR graph flavour
v(t)=6t−2t²
Differentiate → a=6−4t.
Set a=0 → t=1.5 s (max speed).
v(1.5)=4.5 m/s.
Rest → v=0 → t=0 or 3 s.
Sketch the v–t curve: rises, peaks, dips below axis. OCR adores that.
📚 Example 3 – AQA context twist
v(t)=12−2t², s(0)=0
Integrate → s=12t−(2/3)t³.
Rest when v=0 → t²=6 → t=√6.
Return to origin → s=0 → t(12−(2/3)t²)=0 → t²=18 → t=3√2 s.
✅ Add the line “Take t>0 since time cannot be negative.” That phrasing = AQA method mark.
📈 Quick graph cheat sheet
Graph | Gradient gives | Area gives |
s–t | v | — |
v–t | a | displacement |
a–t | — | change in v |
If v–t drops below the axis, that bit’s negative displacement — OCR often tests that by saying “Find total distance.”
❗ Five easy ways to drop marks (and how not to)
- Mixing up distance and displacement → split where v=0.
- Forgetting constants → find them the moment you integrate.
- Using SUVAT out of habit → if a changes, ditch it.
- Skipping units → write m, m/s, m/s² every time.
- Accepting negative time roots → cross them out unless context fits.
🧠 Honestly, these are “one-mark mistakes.” Slow down, tick them off.
📏 Mini formula card
v = ds/dt
a = dv/dt
v = ∫a dt + C
s = ∫v dt + K
Turning point of v → a = 0
Direction change → v = 0
Total distance = Σ |Δs|
Stick that on your notes margin; it’s the whole topic on one napkin.
🧠 Teacher reflection – why students trip up
Right, confession time. Variable acceleration questions rarely beat you with hard maths—they beat you with setup.
People forget an initial condition or sign and boom, the whole structure slides.
Try a “sanity minute” at the end of each question:
1️⃣ Check units.
2️⃣ Check constants.
3️⃣ Check root order.
4️⃣ Check context (time > 0).
5️⃣ Check direction splits.
I swear this 60-second loop has rescued more Edexcel marks than any trick formula.
📊 Little classroom moment
A student once said, “I know how to integrate, but I don’t know when.”
So we drew the ladder: s ↔ v ↔ a.
I asked, “What do you actually have?” She said, “a(t).”
“Cool. Integrate once for v, twice for s.”
Ten seconds later, she grinned — light-bulb moment.
That’s the joy of this topic. Once you see the pattern, it’s just logic wearing calculus clothes.
❓ FAQs
Q 1 – When do I stop using SUVAT?
The second acceleration stops being constant. If a depends on t or s, move to calculus. Write “v=∫a dt, s=∫v dt” and you’ve shown the method.
Q 2 – How do I handle graph-only questions?
Say what you see. Slope = a, area = displacement. OCR and AQA love that wording. Label units and axes clearly.
Q 3 – “Total distance” again — what’s the safe line to write?
“v = 0 at t = …, so split the interval and sum absolute displacements.” It exactly matches the Edexcel scheme.
Q 4 – Can acceleration be negative?
Yep. It just means the object’s slowing or reversing. Describe it in words if asked — AQA markers love that clarity.
🧾 Quick recap table
Concept | Action | Common trap | Fix |
Variable a(t) | Integrate twice → s | Dropped constant | Find immediately |
Turning point of v | a = 0 | Didn’t test sign | Check values around root |
Direction change | v = 0 | Used signed sum | Add absolute values |
Graph question | Confused area/grad | Wrote no units | State m, m/s, m/s² |
Time roots | Negative time | Reject if non-physical |
🧠 Mini-reflection – what makes it click
When you realise acceleration is just “velocity’s gradient,” and displacement is “area under velocity,” you stop memorising and start seeing motion.
That’s why mechanics feels suddenly elegant.
In class I sometimes tell students: “We’re not solving; we’re translating.”
That mindset shift brings the grade jump.
🚀 Next steps & course links
If this topic still feels slippery, don’t panic—everybody hates it before they like it.
Have a look at:
- A Level Maths Topics Students Struggle With – short fixes for common errors.
- A Level Maths Intensive Revision Course – our step-by-step programme for AQA, Edexcel, OCR with live teacher explanations and exam-board phrasing.
Start your A Level Mechanics revision today with our 3-day A Level Maths course — it makes the tricky bits (like variable a) click through conversation, not memorisation.
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, pull everything together with system of forces.”
