A Level Maths: Forces in Two Dimensions

Alright, so this topic always looks neat in the textbook — until you try it. Then the arrows multiply, the slope tilts the wrong way, and suddenly you’re not sure what’s up or down.

That’s fine. Everyone hits that moment with forces in two dimensions.
It’s one of those Mechanics ideas that’s easy once you see it, but awkward until then.

You’ll see it in all the usual exam setups: a box on a rough plane, a car going down a hill, maybe even a pulley. It’s all the same logic — resolve forces, use F = ma, then a bit of SUVAT if they want speed or distance.

Anyway — let’s walk through a few examples together, and I’ll point out where most people go wrong.

🔙 Previous topic:

“Review Newton’s Laws before resolving individual forces.”

⚙️ A Flat Surface to Start

Picture an 8 kg block being pulled along a rough floor by a 45 N force at a 50° angle. There’s 8 N of friction. Classic.

First thing? Always draw the force diagram. Doesn’t need to be fancy — just enough arrows so you know what’s acting where.
Label weight down (mg), reaction up (R), the 45 N slanted line, and friction to the left. That’s it.

Now, go horizontal.
The forward component is 45 cos 50°, and friction drags back with 8 N.

So F = ma becomes:
(45 cos 50° – 8) = 8a → a ≈ 2.6 m/s².

Looks reasonable — not too high, not weirdly small.

From here, if it moves for 5 s from rest, we can get the distance.
Use s = ½ at² → ½ × 2.6 × 25 ≈ 33 m.
Nice tidy number.

✅ Quick thought:
So many students forget to take the horizontal part of the force. Write “horizontal” above your working — it’s an easy mark saver.

⚙️ Let’s Tilt It

Now imagine a 50 kg object sliding down a slope at 65°. Smooth slope, no friction.

Down the slope is where all the action happens, so that’s our positive direction.

Weight acts straight down (50g), but only part of it pulls the object down the slope — that’s 50g sin 65°.

Set up F = ma again:
50g sin 65° = 50a → a ≈ 8.9 m/s².

Pretty quick. The steep slope does the work.

Now, after 20 m of travel, how fast is it going?
We’ll use v² = u² + 2as. Starts from rest, so:
v² = 2 × 8.9 × 20 = 356 → v ≈ 18.9 m/s.

That’s nearly 70 km/h — steep slope indeed.

💡 Note: it’s sin θ for “along the slope” and cos θ for “into the slope.”
If that feels random, think of sin θ as the part that makes it move, cos θ as the part the slope pushes back against.

⚙️ Adding a Bit of Resistance

Alright, last one. A girl on a 55 kg sledge is sliding down a 70° hill. Resistance force = 30 N.
Let’s find acceleration, then her speed after 5 s.

Parallel to the slope, weight pulls down: 55g sin 70°.
Friction pushes up: 30 N.

So, F = ma → 55g sin 70° – 30 = 55a → a ≈ 8.8 m/s².

Decent. You can see the pattern now.

After 5 s: v = u + at → 0 + 8.8 × 5 = 44 m/s.
Fast enough to make you want brakes.

✅ Exam tip:
Always start “parallel to plane” and write it down. Examiners love that phrase — it shows you know what’s going on.

🧩 Let’s Pause

So, what have we actually done here?
Flat surface → slope → slope with resistance.

Same method every time:

Sketch it (forces clear).
Pick a direction.
Resolve the components.
Apply F = ma.
Use SUVAT if needed.

That’s the rhythm — once you’ve done it a few times, you’ll spot the pattern before you even pick up the pen.

💡 Quick Recap

Concept	What to Remember
Resolving forces	Split weight into mg sin θ (along) and mg cos θ (into plane)
F = ma	Always apply parallel to motion
SUVAT	For motion after you find a
Force diagrams	Simple lines, but always draw them

🧠 Final Reflection

You’d be surprised how much smoother Mechanics feels once you stop worrying about “2D” and just think in directions.
There’s only ever one direction you care about at a time.

Don’t chase formulas — follow the logic.

🚀 Next Steps

If these examples helped, you’ll love the full 👉 A Level Maths intensive course — it goes through Mechanics topics like slopes, friction, and connected particles in the same talk-it-through style, with worked examples and quick exam reminders.

Author Bio

S. Mahandru • Head of Maths, Exam.tips

S. Mahandru is Head of Maths at Exam.tips. With over 15 years of experience, he simplifies complex calculus topics and provides clear worked examples, strategies, and exam-focused guidance.

🧭 Next topic:

“Now, discover how two connected objects interact.”

A Level Maths: Forces in Two Dimensions