IB Physics HLTool 3 — MathematicsPractical Skillsalgebra, vectors, units & uncertainties~20 min read
Maths Skills for Physics
Physics is maths with a purpose. Every law you meet is really an equation waiting to be rearranged, every measurement comes with an uncertainty, and every quantity has a unit that has to make sense. This page is your one-stop toolkit for the maths that runs through the whole course: rearranging equations, scalars vs vectors, combining and resolving vectors, units and prefixes, uncertainties, and reading graphs. Get these solid and the rest of the course gets a whole lot easier.
📚 What you need to know
You must select, rearrange, and derive equations to solve problems
A scalar has magnitude only; a vector has magnitude and direction
Perpendicular vectors combine with Pythagoras and are resolved with trigonometry
All units come from 7 SI base units; prefixes (k, m, µ…) scale them by powers of 10
Uncertainty is a range (±) around a measurement — it is not the same as an error
Uncertainties propagate: add absolutes when adding, add fractions when multiplying
A graph’s gradient and area often equal a real physical quantity
Rearranging and using equations
The single most-used skill in physics is taking an equation from the data booklet and bending it to your needs. The good news: you are not expected to memorise the equations — they’re all in the booklet. What you are tested on is picking the right one, rearranging it, and substituting numbers cleanly.
Here’s the reliable recipe for any calculation question:
⚛ Solving a calculation question
List the known quantities with their symbols and units.
Identify the equation in the data booklet that links them.
Rearrange for the quantity you want (before putting numbers in).
Substitute the values and calculate.
Check the units and significant figures of your answer.
Rearrange first, substitute second. Students who plug numbers in too early end up juggling messy decimals and lose track of what they’re solving for. Keep it in symbols until the very last step — it’s cleaner, faster, and far easier to spot mistakes.
WE 1
A car accelerates uniformly. Using v = u + at, find the time taken to reach v = 25 m s−1 from u = 5.0 m s−1 with acceleration a = 2.5 m s−2.
Step 1 — rearrange for tv − u = at, so t = (v − u) / aStep 2 — substitutet = (25 − 5.0) / 2.5 = 20 / 2.5t = 8.0 sNotice the rearranging was done fully in symbols before any numbers went in. That’s the habit to build.
Scalars and vectors
Before you combine any quantities, you have to know what type they are. This one distinction quietly decides half the mistakes in physics.
A scalar has magnitude only — a size, no direction. Examples: mass, time, energy, speed, temperature.
A vector has magnitude and direction. Examples: displacement, velocity, acceleration, force, momentum.
Here’s the trick for telling them apart: ask “can this quantity be negative, meaning it points the other way?” You can have negative displacement (backwards) — that’s a vector. You can’t really have negative energy or negative mass — those are scalars. It’s a quick gut-check that works almost every time.
Scalars (magnitude only)
Vectors (magnitude + direction)
distance
displacement
speed
velocity
mass
acceleration
energy
force
time, temperature
momentum
A vector is drawn as an arrow: the arrow’s length shows the magnitude and the arrowhead shows the direction. That simple picture is the whole basis of adding and resolving vectors, which come next.
Combining and resolving vectors
Two vectors acting together can be replaced by a single resultant vector that has the same overall effect. Going the other way — splitting one vector into two perpendicular parts — is called resolving.
Combining perpendicular vectors
When two vectors meet at right angles, you link them head-to-tail to form a right-angled triangle. The resultant is the hypotenuse, so its magnitude comes from Pythagoras and its direction from trigonometry.
Resultant of two perpendicular vectorsR = √(a² + b²)tanθ = b / a
Link the vectors head-to-tail; the resultant R is the hypotenuse, found with Pythagoras, and its direction θ comes from trigonometry.
WE 2
A drone flies 9.0 km due east, then 12.0 km due north. Find the magnitude and direction (from east) of its displacement.
Step 1 — magnitude with PythagorasR = √(9.0² + 12.0²) = √225 = 15.0 km
Step 2 — direction with trig
θ = tan−1(12.0 / 9.0) = 53°15.0 km, 53° north of eastAlways give a vector answer as BOTH magnitude and direction — a magnitude on its own is only half the answer.
Resolving a vector into components
The reverse move is splitting one vector into a horizontal and a vertical part. If a vector F makes an angle θ with the horizontal, its components are found with the sine and cosine of that angle.
Components of a vectorhorizontal: Fx = F cosθvertical: Fy = F sinθ
A single vector F is equivalent to its horizontal part (F cosθ) plus its vertical part (F sinθ) acting together.
Unsure whether it’s sin or cos? The cos always goes with the angle you’re measuring from — it’s the side lying next to θ (the adjacent side). Picture a “cos sandwich”: the flat, horizontal layer is the cos component. Get that anchored and the sin component is just the other one.
WE 3
A rope pulls a sledge with a force of 8.0 N at 35° above the horizontal. Find the horizontal and vertical components of the force.
Step 1 — horizontal (cos)Fx = 8.0 × cos35° = 6.6 NStep 2 — vertical (sin)Fy = 8.0 × sin35° = 4.6 NFx = 6.6 N, Fy = 4.6 NQuick sanity check: √(6.6² + 4.6²) ≈ 8.0 N, which is the original force. Components should always rebuild the original.
Units and prefixes
Every measurement in physics is built from just seven SI base units. Everything else — the newton, the joule, the volt — is a derived unit made by combining these.
Quantity
Unit
Symbol
length
metre
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
temperature
kelvin
K
amount of substance
mole
mol
To handle very big or very small numbers, we attach a prefix that stands for a power of 10. These are all in the data booklet, but the common ones are worth knowing on sight.
Prefix
Symbol
Value
giga
G
109
mega
M
106
kilo
k
103
centi
c
10−2
milli
m
10−3
micro
µ
10−6
nano
n
10−9
Deriving a unit is just following its definition. For example, force is mass × acceleration, so the newton is:
Deriving the newtonN = kg × m s−2 = kg m s−2
A neat trick that saves you in the exam: check the units on both sides of an equation match. This is called dimensional analysis. If one side comes out in kg m s−2 and the other doesn’t, you know you’ve slipped up somewhere — before you’ve lost a single mark on the numbers.
Uncertainties
No measurement is ever perfect. An uncertainty is the range around a reading within which the true value is expected to lie, written with a ± sign. It is not the same as an error — an error is a mistake or a flaw in the equipment; an uncertainty is an honest estimate of how much wiggle room your reading has.
Uncertainty comes in three flavours, and you should be able to switch between them:
When you put uncertain measurements through a calculation, the uncertainty spreads too. Three simple rules cover almost everything:
Operation
Rule
Adding or subtracting
Add the absolute uncertainties
Multiplying or dividing
Add the fractional (or %) uncertainties
Raising to a power n
Multiply the fractional uncertainty by n
WE 4
A trolley travels s = 36.0 ± 0.2 m in t = 4.00 ± 0.05 s. Find its speed and the absolute uncertainty in that speed.
Step 1 — the speedv = s / t = 36.0 / 4.00 = 9.0 m s−1Step 2 — add fractional uncertainties (division)
0.2/36.0 + 0.05/4.00 = 0.0056 + 0.0125 = 0.018Step 3 — convert back to absolute
Δv = 9.0 × 0.018 = 0.16 m s−1v = 9.0 ± 0.2 m s−1For a divide, add the FRACTIONS, then multiply by the answer to turn it back into an absolute ± value.
WE 5
A sphere has radius r = 2.50 ± 0.02 cm. Its volume is V = &frac43;πr³. Find the volume and its percentage uncertainty.
Step 1 — the volumeV = &frac43;π(2.50)³ = 65.5 cm³Step 2 — power rule (cube → ×3)
% unc in V = 3 × (0.02 / 2.50) × 100 = 2.4%V = 65.5 cm³ ± 2.4%The uncertainty in π is taken as zero, so it doesn’t add anything. Cubing the radius triples its percentage uncertainty — that’s the power rule at work.
💡 Top tips
Rearrange in symbols first, substitute numbers last.
A vector answer needs magnitude AND direction — never just one.
cos pairs with the angle you measure from; sin is the other component.
Uncertainties are usually quoted to 1 significant figure.
Match your answer’s decimal places to its absolute uncertainty.
⚠ Common mistakes
Adding absolute uncertainties when multiplying (you add fractional ones)
Confusing an error (a flaw) with an uncertainty (an estimated range)
Forgetting the direction half of a vector answer
Mixing up sin and cos when resolving — check against the angle
Quoting an answer to more significant figures than the data allows
Reading graphs
Graphs turn a table of numbers into a picture you can actually reason with. Two features matter most: the gradient and the area under the line — each often equals a real physical quantity.
On a straight-line graph of y against x, the gradient is constant and equal to:
Gradient of a straight linem = Δy / Δx = (y2 − y1) / (x2 − x1)
Draw the gradient triangle as large as possible (to reduce error), read the run and rise off the axes, then divide rise by run.
WE 6
On a graph of resistance against length, a straight line passes through (0.3 m, 5.0 Ω) and (1.7 m, 27.0 Ω). Find the gradient.
Step 1 — pick the two points
(x1, y1) = (0.3, 5.0), (x2, y2) = (1.7, 27.0)
Step 2 — rise over runm = (27.0 − 5.0) / (1.7 − 0.3) = 22.0 / 1.4gradient = 15.7 Ω m−1Use points ON the line of best fit, not raw data points that lie off it. And make your triangle big — small triangles magnify reading errors.
The other key feature is the area under a graph. For a velocity–time graph, for example, the area equals the displacement. If the shape is a rectangle or triangle, just use area = base × height (or ½ base × height). For a curve, split it into strips and add them up.
Key idea: on a y–x graph, the gradient tells you the rate of change (how fast y changes with x), and the area tells you the accumulated total — both are usually real physics quantities worth reading off.
Rearrange the equation
substitute & solve
Combine or resolve vectors
track the uncertainty
Read the graph
Quick recap:Rearrange equations in symbols before substituting. Scalars have size only; vectors also have direction. Combine perpendicular vectors with Pythagoras, resolve them with sin/cos. All units come from 7 SI base units scaled by prefixes. Track uncertainties (add absolutes when adding, fractions when multiplying, ×n for powers). Read a graph’s gradient and area as real physical quantities.
These maths skills aren’t a separate topic you learn once and forget — they’re the toolkit you’ll reach for in every single unit, from motion and forces right through to fields and nuclear physics. Whenever a problem feels stuck, come back to the basics: what are my knowns, which equation links them, and have I kept track of units and uncertainties? Next up, you’ll put these tools straight to work in Motion, Forces & Energy.
Maths for physics still feels shaky?
Book a free meeting and we’ll drill rearranging equations, resolving vectors with confidence, and getting uncertainties right every time — the skills that quietly earn marks across the whole course.
