📝 Lesson 1: Variables & Expressions

The language of algebra — translating physics problems into math

🎯 Learning Objectives

Distinguish between constants, variables, and coefficients
Translate word phrases into algebraic expressions
Evaluate expressions by substituting values for variables
Apply order of operations (PEMDAS) correctly

🔑 Key Concepts

Variables vs. Constants

A variable is a symbol (usually a letter) that represents a quantity that can change. In physics, variables represent measurable quantities: v for velocity, t for time, m for mass, F for force.

A constant is a fixed value that doesn't change. In F = ma, the mass m might be constant for a given problem, while F and a vary.

A coefficient is the number multiplied by a variable. In 3x, the coefficient is 3. In physics formulas, coefficients often have physical meaning (like ½ in KE = ½mv²).

Translating Words to Algebra

Phrase	Algebra	Physics Example
"5 more than x"	`x + 5`	Position: `x = x₀ + 5`
"3 times a number"	`3n`	Force: `F = 3m` (if a = 3)
"half of v"	`v/2` or `½v`	Average velocity: `v_avg = (v₁ + v₂)/2`
"the sum of a and b"	`a + b`	Total momentum: `p_total = p₁ + p₂`
"the difference of x and y"	`x - y`	Displacement: `Δx = x_f - x_i`
"t squared"	`t²`	Kinematics: `x = ½at²`

Order of Operations (PEMDAS)

Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right)

Physics Pro Tip: In physics, parentheses often group terms with physical meaning — like (v_f - v_i) for change in velocity. Never skip them!

Evaluating Expressions

Substitute the given values for each variable, then simplify using PEMDAS.

Example: Evaluate ½mv² when m = 2 kg and v = 10 m/s.

½(2)(10)²
= ½(2)(100)        ← Exponent first
= 1(100)            ← Multiply: ½ × 2 = 1
= 100 J             ← Kinetic energy in joules

📐 Worked Examples

Example 1: Translating a Physics Statement

Problem: "The final velocity equals the initial velocity plus acceleration times time."

Solution: Identify variables: v_f (final velocity), v_i (initial velocity), a (acceleration), t (time).

v_f = v_i + a·t

Example 2: Evaluating with Units

Problem: A 5 kg object moves at 8 m/s. Find its kinetic energy using KE = ½mv².

KE = ½(5 kg)(8 m/s)²
   = ½(5 kg)(64 m²/s²)
   = 2.5 kg × 64 m²/s²
   = 160 kg·m²/s²
   = 160 J

Example 3: Nested Parentheses

Problem: Simplify 3[2(x + 4) - 5] when x = 3.

3[2(3 + 4) - 5]
= 3[2(7) - 5]       ← Inner parentheses first
= 3[14 - 5]         ← Multiply
= 3[9]              ← Subtract
= 27                ← Multiply

🏋️ Practice Problems

Translate to Algebra:

The force equals mass times acceleration.
The displacement equals average velocity times time.
Potential energy equals mass times gravity times height.
The net force equals the sum of all individual forces.
Frequency equals one divided by period.

Evaluate (show your work):

v_f = v_i + at when v_i = 5 m/s, a = 2 m/s², t = 3 s
KE = ½mv² when m = 10 kg, v = 6 m/s
F = G(m₁m₂)/r² when m₁ = m₂ = 1000 kg, r = 10 m, G = 6.67×10⁻¹¹
x = x₀ + v₀t + ½at² when x₀ = 0, v₀ = 4 m/s, a = -2 m/s², t = 5 s

Simplify:

2(x + 3) - 4(x - 1)
½(2m + 4m)v²
(F₁ + F₂)/m when F₁ = 3ma and F₂ = ma

✅ Answers

Click to reveal answers

Translate: 1) F = ma 2) Δx = v_avg·t 3) PE = mgh 4) F_net = ΣF 5) f = 1/T

Evaluate: 6) 11 m/s 7) 180 J 8) 6.67×10⁻⁵ N 9) 5 m

Simplify: 10) -2x + 10 11) 3mv² 12) 4a

← Back to Algebra Home Next: Solving Linear Equations →