Algebra Essentials
for Physics
Sharpen your algebraic toolkit: rearrange any formula, solve equations confidently, and manipulate physical relationships like a physicist — not a textbook robot.
🎯 Learning Outcomes
- Rearrange multi-variable physics formulas for any unknown
- Solve linear, simultaneous, and quadratic equations
- Work with powers, roots, and fractional exponents
- Apply the substitution method to solve physics problems
- Use proportionality and inverse relationships correctly
- Recognise common algebraic traps in physics problem-solving
📋 Prerequisites
- Lesson 1.1 – Numbers, Units & Dimensional Analysis
- Basic arithmetic (fractions, decimals, BODMAS)
- Understanding of variables and expressions
- No calculus required here
01 The Concept
Why Algebra Runs Physics
Every physics equation is a relationship between quantities. The real skill isn't memorising formulas — it's being able to rearrange them to find whatever quantity you need. When you know $F = ma$, you automatically know how to find $m$ if you're given $F$ and $a$. That flexibility comes from algebra.
In physics problems you'll constantly need to: isolate an unknown, substitute one equation into another, handle powers and roots, and check your working using units. This lesson drills those skills.
1.1 Rearranging Formulas — the Golden Rule
The golden rule: whatever operation you do to one side of an equation, you must do to the other. The equals sign is a perfect balance — never disturb one side without balancing the other.
🔑 Step-by-Step Strategy for Rearranging
- Identify your target variable (what you want to isolate)
- Move everything else to the other side using inverse operations
- Inverse operations: + ↔ −, × ↔ ÷, power ↔ root
- Check your answer dimensionally — units must balance
Common Rearrangements You'll Encounter
| Original | Rearranged for… | Result |
|---|---|---|
| $v = u + at$ | acceleration $a$ | $a = \dfrac{v - u}{t}$ |
| $v = u + at$ | time $t$ | $t = \dfrac{v - u}{a}$ |
| $F = ma$ | mass $m$ | $m = \dfrac{F}{a}$ |
| $E_k = \tfrac{1}{2}mv^2$ | velocity $v$ | $v = \sqrt{\dfrac{2E_k}{m}}$ |
| $T = 2\pi\sqrt{\dfrac{L}{g}}$ | length $L$ | $L = \dfrac{T^2 g}{4\pi^2}$ |
| $v^2 = u^2 + 2as$ | displacement $s$ | $s = \dfrac{v^2 - u^2}{2a}$ |
1.2 Solving Linear Equations
A linear equation has the unknown raised only to the first power. Most introductory physics equations reduce to linear form once you've made a substitution.
The systematic approach: collect terms with the unknown on one side, factor out the unknown if it appears more than once, then divide both sides.
1.3 Simultaneous Equations
When two unknowns appear across two equations, you need simultaneous solution. In physics this crops up constantly: two bodies interacting, equilibrium of forces in two directions, circuit equations.
Two reliable methods:
- Substitution: solve one equation for a variable, substitute into the other
- Elimination: multiply equations to match coefficients, then add/subtract
1.4 Powers, Roots & Fractional Exponents
Physics is full of squares, cubes, and square roots. The key rules:
Add exponents when multiplying same base
Subtract exponents when dividing
Square root = power of ½
Negative power = reciprocal
1.5 Proportionality
Physicists often work with proportional relationships before pinning down exact formulas. Two key types:
- Direct proportion: $y \propto x$ means doubling $x$ doubles $y$. Written $y = kx$.
- Inverse proportion: $y \propto \tfrac{1}{x}$ means doubling $x$ halves $y$. Written $y = \tfrac{k}{x}$.
- Square law: $y \propto x^2$ means doubling $x$ quadruples $y$.
Proportionality reasoning is often faster than full substitution for multiple-choice problems — learn to use it as a check.
1.6 Quadratic Equations in Physics
Kinematic equations under constant acceleration often produce quadratics in time $t$. The standard form is $at^2 + bt + c = 0$, solved by:
Always gives two solutions — pick the physically meaningful one
$\Delta > 0$: two real solutions; $\Delta = 0$: one; $\Delta < 0$: no real solution
02 Key Equations Reference
$m$ = gradient (slope), $c$ = y-intercept
$k$ = constant of proportionality
For $ax^2 + bx + c = 0$
Apply inverse operation (root) to both sides
03 Worked Examples
Question: The kinematic equation is $v^2 = u^2 + 2as$. A car starts from rest ($u = 0$) and reaches $v = 30$ m/s over a distance $s = 45$ m. Find the acceleration $a$.
Question: Two objects A and B have masses $m_A$ and $m_B$. After a collision: $m_A + m_B = 5$ kg and $m_A - m_B = 1$ kg. Find both masses.
Question: A ball is thrown upward from ground level with initial velocity $u = 20$ m/s. Using $s = ut - \tfrac{1}{2}gt^2$ with $g = 10$ m/s², find the time(s) when $s = 15$ m.
Question: Gravitational force between two masses is $F = \dfrac{Gm_1 m_2}{r^2}$. If the distance $r$ is doubled and mass $m_1$ is tripled, what happens to $F$?
04 Practice Problems
The equation for kinetic energy is $E_k = \tfrac{1}{2}mv^2$. Rearrange to make $v$ the subject. Then find $v$ when $E_k = 200$ J and $m = 4$ kg.
Using $F = ma$, find the acceleration of a 1200 kg car when a net force of 3600 N is applied. Then find the force needed to give a 70 kg person an acceleration of 5 m/s².
The period of a pendulum is $T = 2\pi\sqrt{\dfrac{L}{g}}$. A pendulum on Earth ($g = 9.81$ m/s²) has $T = 2$ s. Find its length $L$. Show all algebraic steps.
Solve the simultaneous equations to find velocity components $v_x$ and $v_y$: $$v_x + v_y = 10 \text{ m/s}$$ $$2v_x - v_y = 5 \text{ m/s}$$
A stone is dropped from rest ($u = 0$) and falls under gravity ($g = 10$ m/s²). Using $s = ut + \tfrac{1}{2}gt^2$, find the time when $s = 80$ m. Use the quadratic formula.
The intensity of light from a point source is $I = \dfrac{P}{4\pi r^2}$, where $P$ is power and $r$ is distance. (a) Rearrange for $r$. (b) If intensity drops to $\tfrac{1}{9}$ of its original value, what happened to $r$? (c) A bulb of 60 W produces intensity $I = 0.12$ W/m² at some distance. Find that distance.
Show Full Worked Answers
Rearrange: $v = \sqrt{\dfrac{2E_k}{m}}$
Substituting: $v = \sqrt{\dfrac{2 \times 200}{4}} = \sqrt{100} = \mathbf{10 \text{ m/s}}$
Car: $a = \dfrac{F}{m} = \dfrac{3600}{1200} = \mathbf{3 \text{ m/s}^2}$
Person: $F = ma = 70 \times 5 = \mathbf{350 \text{ N}}$
Step 1 — rearrange for $L$:
$T = 2\pi\sqrt{\dfrac{L}{g}}$ → Square both sides: $T^2 = 4\pi^2 \dfrac{L}{g}$
→ $L = \dfrac{T^2 g}{4\pi^2}$
Step 2 — substitute $T = 2$, $g = 9.81$:
$L = \dfrac{4 \times 9.81}{4\pi^2} = \dfrac{39.24}{39.48} \approx \mathbf{0.993 \text{ m}} \approx 1 \text{ m}$ ✓
Equations: (1) $v_x + v_y = 10$, (2) $2v_x - v_y = 5$
Add (1) + (2): $3v_x = 15 \Rightarrow v_x = 5$ m/s
Substitute into (1): $5 + v_y = 10 \Rightarrow v_y = 5$ m/s
$\mathbf{v_x = v_y = 5 \text{ m/s}}$
With $u = 0$: $80 = \tfrac{1}{2}(10)t^2 \Rightarrow t^2 = 16 \Rightarrow t = \mathbf{4 \text{ s}}$
(No quadratic needed here since $u = 0$, but confirm: $\tfrac{1}{2}(10)(16) = 80$ ✓)
(a) $r = \sqrt{\dfrac{P}{4\pi I}}$
(b) Since $I \propto \dfrac{1}{r^2}$: if $I' = \tfrac{1}{9}I$, then $r'^2 = 9r^2 \Rightarrow r' = 3r$. Distance tripled.
(c) $r = \sqrt{\dfrac{60}{4\pi \times 0.12}} = \sqrt{\dfrac{60}{1.508}} = \sqrt{39.8} \approx \mathbf{6.3 \text{ m}}$
05 Summary & Common Mistakes
📌 Key Takeaways
- Rearrange using inverse operations — always balance both sides
- Verify rearrangements dimensionally before substituting numbers
- Simultaneous equations: use elimination or substitution consistently
- Quadratics in physics — always ask "which solution is physical?"
- $\sqrt{x^2} = |x|$ — don't lose the sign when taking roots
- Proportionality is your speed tool for ratio questions
⚠️ Common Mistakes
- Half the square root: $\sqrt{ab} \ne \tfrac{1}{2}\sqrt{a}\sqrt{b}$ — distribute carefully
- Forgetting $\pm$: $v^2 = 100 \Rightarrow v = \pm 10$ — check both signs
- Wrong inverse: $\tfrac{1}{2}mv^2 = E$ → students write $v = \sqrt{\tfrac{E}{m}}$ missing the 2
- Arithmetic slip after algebra: do algebra first, substitute last
- Discarding negative time: sometimes negative $t$ means "before the clock started" — read the problem
- Treating $\propto$ as $=$: proportionality introduces a constant $k$ — don't drop it
🧠 Physicist's Workflow
Always work symbolically first — rearrange the formula to isolate your unknown before plugging in numbers. This avoids arithmetic errors, makes units easier to check, and often reveals cancellations that simplify the calculation dramatically.
