Math for Physics Study Guide

Level High SchoolReview time ~10 hours

Physics is mathematics describing nature, and a small toolkit — vectors, trig, scientific notation, graphs, basic calculus, and honest error handling — unlocks every subject on this site. This guide sharpens each tool on real physics formulas.

Key formulas

Kinetic Energy Formula — KE = ½mv²

Kinetic energy equals one-half times mass times velocity squared.

Newton's Second Law — F = ma

The net force on an object equals its mass times its acceleration.

Ohm's Law — V = IR

Voltage equals current times resistance.

Wave Equation — v = fλ

Wave speed equals frequency times wavelength.

Topic checklist

Vectors and scalars

Add vectors by components or tip-to-tail; never add magnitudes of non-parallel vectors.
Resolve into $x$ and $y$ components with sine and cosine.
Distinguish vector equations from scalar ones at a glance.

Trigonometry

Use SOH-CAH-TOA to find components and resultants.
Apply the Pythagorean theorem to perpendicular components.
Find direction with $\theta = \tan^{-1}(F_y/F_x)$ , checking the quadrant.

Scientific notation

Write large and small numbers in $a \times 10^n$ form.
Multiply and divide by adding and subtracting exponents.
Estimate orders of magnitude before calculating precisely.

Graphing

Read slopes as rates and areas as accumulated quantities.
Interpret curved graphs by considering tangents and integrals.
Linearize data: plot the right variables to get a straight line.

Calculus connections

Connect derivatives to rates: $v = dx/dt$ , $a = dv/dt$ .
Connect integrals to accumulation: $\Delta x = \int v\,dt$ .
Use $\int F\,dx$ for work with varying forces.

Significant figures and error

Count significant figures in measurements and results.
Round final answers to the least precise input's figures.
Distinguish random from systematic errors and reduce each properly.

Common mistakes

Mistake: Adding vector magnitudes directly.

Fix:

$3\ \mathrm{m}$ east plus $4\ \mathrm{m}$ north is $5\ \mathrm{m}$ at $53°$ — not $7\ \mathrm{m}$ . Components add; magnitudes do not. Draw the triangle every time.

Mistake: Mixing up sine and cosine in components.

Fix:

$\sin\theta$ is opposite over hypotenuse of the angle you defined. Label the angle on your diagram first, then assign sine and cosine — swapped components are the classic error.

Mistake: Exponent arithmetic errors in scientific notation.

Fix:

$10^3 \times 10^5 = 10^8$ , and $3.0 \times 10^8$ has two significant figures. Keep the coefficient between 1 and 10 and track the exponent separately.

Mistake: Reading slope when the question asks for area (or vice versa).

Fix:

On a velocity-time graph, slope is acceleration and area is displacement. Say the words 'slope of' and 'area under' explicitly before reading any value off a graph.

Mistake: Reporting too many significant figures.

Fix:

A result cannot be more precise than its inputs: $12.5 \times 8.24 = 103$ , not $103.0$ . Count the significant figures of each input and round the answer to match the weakest.

Exam tips

Draw the vector triangle before writing any components.
Do a rough order-of-magnitude estimate before the precise calculation — it catches blunders.
Keep track of units through every algebra step; they are free error-checking.
Round only at the end; intermediate rounding poisons the final digits.