IB Physics SL
Tool 3 — Mathematics
Paper 1 & 2
Select · manipulate · derive
~8 min read
Applying General Mathematics in Physics
Physics is maths with a story attached. Behind every calculation question is a short chain of moves: spot the quantities you’re given, find the equation that links them, rearrange it, and put the numbers in. The good news — you don’t have to memorise the equations, they’re in the data booklet. What you’re really being tested on is your ability to choose, rearrange, and combine them. This page gets those moves feeling automatic.
📘 What you need to know
- You’ll apply arithmetic (add, subtract, multiply, divide) to algebraic expressions and rearrange them to isolate a variable
- Common area and volume formulas (rectangle, triangle, circle, cuboid, cylinder, sphere) show up constantly — most are in the data booklet
- Be fluent with decimals, fractions, percentages and ratios, and with reciprocals and exponents
- Know your trigonometric ratios (SOH CAH TOA) for right-angled triangles — vital for vectors
- The calculation recipe: list knowns → pick the equation → rearrange → substitute
- Deriving an equation means combining fundamental relationships to build a new one; a “show that” gives you the target to aim for
- Watch for keyword clues: “stationary” means u = 0, “constant velocity” means net force and acceleration are zero
Algebra and rearranging equations
Solving physics problems with algebra comes down to two skills: applying the ordinary rules of arithmetic to expressions, and manipulating those expressions to get the variable you want on its own. Whatever you do to one side of an equation, you do to the other — that’s the rule that keeps everything balanced.
Say you’re given the equation of motion v = u + at and asked to make time the subject. Peel the equation apart one operation at a time:
Rearranging for t
v − u = at → t = (v − u) ÷ a
Think of the variable you want as a parcel wrapped in layers. To unwrap t, you reverse whatever was done to it, working from the outside in: first move the u across, then undo the multiplication by a. Do the operations in reverse order and you can’t go wrong.
Areas, volumes and other shape formulas
Geometry sneaks into physics all the time — the cross-section of a wire, the volume of a sphere of gas, the area a pressure acts on. Most of these are handed to you in the data booklet, so the skill is recognising which one you need and substituting correctly.
- Areas: rectangle A = lw; triangle A = ½bh; circle A = πr2
- Volumes: cuboid V = lwh; cylinder V = πr2h; sphere V = (4÷3)πr3
- Surfaces: circumference C = 2πr; sphere surface A = 4πr2
These combine with definitions to solve real problems. Density is mass per unit volume, so for a cylindrical wire of mass m, radius r and length L, substitute the cylinder volume straight into the density formula:
Density of a wire
ρ = m ÷ V = m ÷ (πr2L)
Fractions, percentages, reciprocals and exponents
A few everyday number skills carry most of the load in physics. Fractions turn up in algebra and uncertainty work — and remember the S⇔D button on your calculator flips a fraction to a decimal. Percentages appear as percentage change, difference, error and uncertainty. Ratios let you compare quantities cleanly.
Two more worth naming outright. A reciprocal is just one divided by the number, which is why period and frequency are reciprocals of each other: T = 1 ÷ f. An exponent is a power a number is raised to — as in the SHM acceleration relationship a = −ω2x.
Trigonometric ratios
Right-angled triangles are everywhere in physics, especially when you resolve vectors into components. The three ratios link an angle to the sides of the triangle:
SOH CAH TOA
sin θ = O ÷ H · cos θ = A ÷ H · tan θ = O ÷ A
Quick recap: rearrange by reversing operations, pull the right shape or equation from the data booklet, and keep SOH CAH TOA ready for anything with an angle.
The calculation recipe
Almost every “calculate” question yields to the same four-step routine. Learn it once and it works whatever the topic.
🧭 Solving a calculation question
- List the known quantities — write each one with its symbol and unit so nothing hides
- Identify the equation in the data booklet that connects them
- Rearrange it for the quantity you want, if it isn’t already the subject
- Substitute the values and compute the final answer with its unit
List knowns
→
Pick equation
→
Rearrange
→
Substitute
WE 1A go-kart travelling at constant velocity covers 45 m in 3.0 s. The driving force from the engine is 1.2 kN. Calculate the power output of the go-kart.
Two power equations exist: P = ΔW ÷ Δt and P = Fv. We’re given force, distance and time, so combine the first with work done W = Fs.
List & select
s = 45 m, t = 3.0 s, F = 1.2 kN = 1200 N
P = Fs ÷ t
Substitute
P = (1200 × 45) ÷ 3.0
P = 18 000 W = 18 kW
Same answer via P = Fv, since v = s/t = 15 m s⁻¹ and 1200 × 15 = 18 000 W.
Deriving relationships
Deriving means building a new equation by combining ones you already trust. Start by naming the fundamental principle at play — a force balance, energy conservation, momentum — then list the relevant equations and manipulate them together. If it’s a “show that” question, the target equation is printed for you, so you just steer the algebra towards it.
WE 2A copper cylinder of cross-sectional area A and length L has resistance 12 Ω. A second copper cylinder has twice the area (2A) and three times the length (3L). Find its resistance.
Resistivity gives ρ = RA ÷ L, so R = ρL ÷ A. Since ρ is the same for both, write it as a ratio to cancel the constant.
As a proportion
R ∝ L ÷ A
R₂ ÷ R₁ = (L₂ ÷ L₁) × (A₁ ÷ A₂)
Substitute the changes
R₂ ÷ R₁ = 3 × (1 ÷ 2) = 1.5
R₂ = 1.5 × 12
R₂ = 18 Ω
Tripling length raises R; doubling area lowers it — the net effect is a 1.5× increase.
💡 Top tips
- Always list knowns with units first. It’s the fastest way to spot which data-booklet equation fits
- Rearrange by reversing operations from the outside in — undo addition before you undo multiplication
- Carry at least two extra significant figures through working, and only round the final answer
- Decode the keywords: “stationary” ⇒ u = 0; “constant velocity” ⇒ net force = 0 and a = 0; “rebounds” ⇒ the velocity sign flips
⚠ Common mistakes
- Forgetting to convert prefixes before substituting — 1.2 kN is 1200 N, not 1.2
- Rearranging in the wrong order, so a variable gets stranded under a division or inside a bracket
- Trying to memorise equations instead of practising how to select and manipulate the ones provided
- In a ratio question, inverting a factor — larger area lowers resistance, so A goes on the bottom of R ∝ L/A
Up next: Scalar & Vector Quantities — where these trig ratios and rearranging skills start doing real work, splitting forces and displacements into components.
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