🔧 Lesson 3: Formula Manipulation (Literal Equations)

The #1 skill for physics — solving any formula for any variable

🎯 Learning Objectives

Solve literal equations (equations with multiple variables) for any specified variable
Recognize that the same algebraic steps apply whether variables are numbers or letters
Rearrange common physics formulas: v = d/t, F = ma, KE = ½mv², p = mv, etc.
Use formula manipulation to derive new relationships

🔑 Key Concepts

What Is a Literal Equation?

A literal equation is an equation where variables represent known quantities — like formulas in physics. The letters aren't unknowns to solve for once; they're placeholders for quantities you might need to find in different situations.

🚨 Critical: In physics, you'll be given a formula like v = d/t and asked to find d when you know v and t. You MUST be able to rearrange to d = vt instantly. This is not optional — it's the gateway skill.

The Process Is Identical

Solving 3x + 5 = 14 for x uses the exact same steps as solving v = d/t for d:

Numeric Equation	Literal Equation
3x + 5 = 14	v = d/t
3x = 14 - 5	v·t = d (multiply both sides by t)
x = 9/3 = 3	d = vt (symmetric property)

Common Physics Formulas — Know All Rearrangements

Formula	Solve for Each Variable
`v = d/t`	`d = vt` \| `t = d/v`
`a = Δv/t`	`Δv = at` \| `t = Δv/a`
`F = ma`	`m = F/a` \| `a = F/m`
`p = mv`	`m = p/v` \| `v = p/m`
`KE = ½mv²`	`m = 2KE/v²` \| `v = √(2KE/m)`
`PE = mgh`	`m = PE/gh` \| `h = PE/mg` \| `g = PE/mh`
`F = kx` (Hooke's Law)	`k = F/x` \| `x = F/k`
`v_f = v_i + at`	`v_i = v_f - at` \| `a = (v_f - v_i)/t` \| `t = (v_f - v_i)/a`
`x = x₀ + v₀t + ½at²`	Practice rearranging for each variable!

📐 Worked Examples

Example 1: Kinematics — Solve for Initial Velocity

Formula: v_f = v_i + at. Solve for v_i.

v_f = v_i + at
v_f - at = v_i       ← Subtract at from both sides
v_i = v_f - at       ← Rewrite with target variable on left

Example 2: Energy — Solve for Velocity

Formula: KE = ½mv². Solve for v.

KE = ½mv²
2·KE = mv²           ← Multiply both sides by 2
2KE/m = v²           ← Divide both sides by m
v = √(2KE/m)         ← Take square root (v > 0 for speed)

Physics Note: When taking square roots in physics, consider whether the quantity can be negative. Speed is always positive; velocity can be negative (direction). Use ±√ for velocity, √ for speed.

Example 3: Gravitation — Solve for Distance

Formula: F = Gm₁m₂/r². Solve for r.

F = Gm₁m₂/r²
Fr² = Gm₁m₂          ← Multiply both sides by r²
r² = Gm₁m₂/F         ← Divide both sides by F
r = √(Gm₁m₂/F)       ← Take square root (distance > 0)

Example 4: Thin Lens — Solve for Image Distance

Formula: 1/f = 1/do + 1/di. Solve for di.

1/f = 1/do + 1/di
1/di = 1/f - 1/do    ← Subtract 1/do from both sides
1/di = (do - f)/(f·do)  ← Common denominator
di = f·do/(do - f)   ← Take reciprocal (flip both sides)

🏋️ Practice Problems

Rearrange each formula for the indicated variable:

a = (v_f - v_i)/t → solve for v_f, then v_i, then t
F = Gm₁m₂/r² → solve for m₁, then r
p = mv → solve for m, then v
KE = ½mv² → solve for m, then v
PE = mgh → solve for h, then g
v = fλ (wave speed) → solve for f, then λ
F = kx → solve for k, then x
x = x₀ + v₀t + ½at² → solve for v₀, then a (harder!)
v_f² = v_i² + 2aΔx → solve for v_i, then a, then Δx
1/f = 1/do + 1/di → solve for do (challenge!)

✅ Answers

Click to reveal answers

1) v_f = v_i + at; v_i = v_f - at; t = (v_f - v_i)/a

2) m₁ = Fr²/(Gm₂); r = √(Gm₁m₂/F)

3) m = p/v; v = p/m

4) m = 2KE/v²; v = √(2KE/m)

5) h = PE/mg; g = PE/mh

6) f = v/λ; λ = v/f

7) k = F/x; x = F/k

8) v₀ = (x - x₀ - ½at²)/t; a = 2(x - x₀ - v₀t)/t²

9) v_i = √(v_f² - 2aΔx); a = (v_f² - v_i²)/(2Δx); Δx = (v_f² - v_i²)/(2a)

10) do = f·di/(di - f)

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