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Rearranging Equations with Symbols

Symbols stand for quantities with units, subscripts are labels and Δ means final minus initial.

Practice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

Physics equations look like algebra with the wrong letters. They are the same algebra you already know: x is not a mystery number but a position in meters, and t is a time in seconds. Once you read the symbols as quantities, rearranging an equation is just undoing what was done, step by step.

Part 2 · Prerequisite check

Quick check before you start

1. What is the SI unit of velocity?

  1. m/s
  2. km/h
  3. m
  4. s/m
Show the answer

Velocity is a displacement in meters per time in seconds.

  • Correct: m/s:
  • km/h:
  • m:
  • s/m:

2. A ball moves from x = 2.0 m to x = 5.0 m. What is its displacement?

  1. +3.0 m
  2. +7.0 m
  3. −3.0 m
  4. +5.0 m
Show the answer

Displacement is final position minus starting position: 5.0 m − 2.0 m = +3.0 m.

  • Correct: +3.0 m:
  • +7.0 m:
  • −3.0 m:
  • +5.0 m:

Part 3 · See it

See it first

The equation x = x-zero plus v t, with callouts: x is the position at time t in meters; x-zero, with subscript 0, is the starting position at t = 0; v is the velocity in meters per second; t is the time since the start in seconds. Below, a two-column rearrangement solves for t: x = x0 + vt; subtract x0 from both sides to get x − x0 = vt; divide both sides by v to get (x − x0)/v = t; write x − x0 as delta x, so t = delta x over v.
Each symbol is a quantity with a unit. To solve for t, undo the addition first, then the multiplication, doing the same to both sides. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

  1. Each letter in an equation stands for a quantity with a unityou can check any rearranged equation by checking its units
  2. An equation stays true when you do the same thing to both sidesyou can isolate any symbol by undoing operations in reverse order
  3. You solve in symbols before putting in numbersthe result shows how the answer depends on each quantity, and you substitute only once
  4. Δ means final minus initiala change can be negative, and its sign carries the direction

Part 5 · Key ideas

Key ideas

  • A symbol is a quantity with a unit: x position (m), v velocity (m/s), t time (s). Subscripts are labels: x₀ is the starting position.
  • Δ means "change in" and is always final minus initial: Δx = x − x₀. A negative Δ means the quantity went down.
  • To rearrange, undo operations in reverse order, doing the same to both sides: from x = x₀ + vt, subtract x₀, then divide by v.
  • Solve in symbols first, then substitute numbers with units once. Check that the units of the result make sense.

Part 6 · Misconception

A common mistake

The wrong idea: To solve d = vt for t, divide v by d, because t is "on the other side" of v.

What actually happens: Divide both sides by v: t = d/v. A units check catches the slip: d/v is m ÷ (m/s) = s, while v/d is 1/s.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. Solve d = vt for t.

  1. t = v / d
  2. t = d / v
  3. t = dv
  4. t = d − v
Show the answer

d = vt. Divide both sides by v: d/v = t. Units check: m ÷ (m/s) = s.

  • t = v / d: This flips the fraction. Check with units: (m/s) ÷ m = 1/s, not seconds.
  • Correct: t = d / v: Right: divide both sides by v, and v cancels on the right.
  • t = dv: Multiplying by v does not undo multiplying by v; divide instead.
  • t = d − v: v multiplies t, so it is removed by dividing, not subtracting.

2. Solve x = x₀ + vt for v.

  1. v = (x − x₀) / t
  2. v = x / t − x₀
  3. v = (x + x₀) / t
  4. v = t / (x − x₀)
Show the answer

Undo in reverse order: x − x₀ = vt, then v = (x − x₀)/t.

  • Correct: v = (x − x₀) / t: Right: subtract x₀ from both sides, then divide both sides by t.
  • v = x / t − x₀: Dividing first leaves x₀/t, not x₀. Undo the addition before the multiplication.
  • v = (x + x₀) / t: x₀ was added on the right, so it is removed by subtracting it from both sides.
  • v = t / (x − x₀): The fraction is upside down; check units: s ÷ m is not m/s.

3. A thermometer reads 14.5 °C at noon and 9.0 °C at 6 p.m. What is ΔT, the change in temperature? Give a signed value.

Type a number in °C.

Show the answer

Δ always means final minus initial: ΔT = 9.0 °C − 14.5 °C = −5.5 °C. The minus sign says the temperature went down.

  • Answer: -5.5 °C

4. A rolling ball starts at x₀ = 1.20 m and moves at a steady v = 0.45 m/s. Using x = x₀ + vt, where is it at t = 4.0 s?

Type a number in m.

Show the answer

Substitute with units: x = 1.20 m + (0.45 m/s)(4.0 s) = 1.20 m + 1.8 m = 3.0 m. 0.45 and 4.0 each have two significant figures.

  • Answer: 3.0 m

5. A student derives t = v / d for the time to cover distance d at speed v. Which check shows the expression is wrong?

  1. Its units are (m/s) ÷ m = 1/s, not seconds
  2. It contains two symbols instead of one number
  3. It does not include the starting position x₀
  4. It does not depend on the distance d at all
Show the answer

Units check: (m/s)/m = 1/s. A time must be in s, so the expression is upside down; t = d/v gives m ÷ (m/s) = s. It also gives less time for a faster speed, as it should.

  • Correct: Its units are (m/s) ÷ m = 1/s, not seconds: Right: a correct expression for a time must come out in seconds. Checking units catches upside-down fractions.
  • It contains two symbols instead of one number: A symbolic answer is supposed to contain symbols; that is not the problem.
  • It does not include the starting position x₀: The question asks only for a time over a distance; x₀ is not needed.
  • It does not depend on the distance d at all: d is in the expression; the time does depend on it, but upside down, which is the real problem.

6. Solve x = x₀ + vt for t, then find t when x = 9.0 m, x₀ = 3.0 m and v = 1.5 m/s.

Type a number in s.

Show the answer

t = (x − x₀)/v = (9.0 m − 3.0 m) ÷ 1.5 m/s = 4.0 s. Rearrange in symbols first, then substitute once.

  • Answer: 4.0 s

Part 8 · Summary

Summary

Symbols stand for quantities with units, subscripts are labels and Δ means final minus initial. Rearrange by undoing operations in reverse order on both sides, solve in symbols first, then substitute numbers with units and check the units of the result.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections