An equation is just a sentence
Physics is written in equations, and that scares a lot of people away. So let's start by taking the fear out of it: an equation is a sentence written in shorthand.
Here is a normal sentence:
“The distance you travel equals your speed multiplied by the time you spend travelling.”
And here is exactly the same sentence written the way a physicist would:
Each letter is a variableA symbol (usually a letter) that stands for a quantity whose value can change — like the speed of a car.: a short name for a quantity. is distance, is speed (we'll call it velocity later), and is time. Physicists use single letters only because they get tired of writing the whole word.
The = sign means 'balanced'
The most important symbol in all of physics is the humble equals sign. It doesn't mean “the answer is coming”. It means: the thing on the left and the thing on the right are the same amount — just written in two different ways.
That's why the perfect picture of an equation is an old-fashioned balance scale. If the two sides weigh the same, the beam is level, and the equation is true. Below is the equation as a scale. Each purple box weighs some unknown amount . Your job: find out how much one box weighs, without ever letting the scale tip.
A balance scale holding an equation, with weights for x on one side and numbers on the other. Use the buttons to do the same operation to both sides until x is alone; the scale stays level whenever the two sides are equal.
The equation balance
InteractiveThe math, with your numbers
Each purple box weighs the same unknown amount x. Each orange block weighs 1.
Try this
- →Solve the puzzle using only moves that keep the scale level.
- →Remove a weight from the left side only. What happens to the 'equals' sign?
- →Try a new puzzle. Can you always do 'remove the loose weights' first, then 'split into groups'?
Undo moves: every operation has an opposite
To get a letter by itself (that's what “solving for x” means) you peel away everything around it using the opposite operation:
Worked example: solve 2x + 3 = 11
Step 1 of 5- 1Start with the equation. The x is trapped behind a '× 2' and a '+ 3'.
Doing it with a real physics formula
Here's the payoff. Physics formulas are solved exactly the same way — the letters just stand for real things.
Say it out loud: “distance equals speed times time.”
Distance· measured in metres (m)
How far you went.
Speed· measured in metres per second (m/s)
How many metres you cover each second.
Time· measured in seconds (s)
How long you were moving.
Make a prediction first
You ran 100 metres in 20 seconds. The formula gives distance, but you want the speed. What do you do?
Think about it, then click to reveal the answer.
Make a prediction first
You ran 100 metres in 20 seconds. The formula gives distance, but you want the speed. What do you do?
Think about it, then click to reveal the answer.
Get by itself. Right now it's being multiplied by , so divide both sides by :
You ran 5 metres every second. Same balance rule — nothing new.
Decoding the strange symbols
Physics borrows letters from the Greek alphabet, and a few marks with special meanings. You only need a handful for this whole course:
| Delta | "change in". Δt = (time at the end) − (time at the start). | |
| theta | an angle. | |
| mu | how 'grippy' two surfaces are (friction). | |
| omega | how fast something spins or oscillates. | |
| lambda | the length of one wave. | |
| Sigma | "add them all up". | |
| v-naught | a little 0 underneath means 'at the start'. v₀ = starting speed. |
Wrap-up
Key ideas to remember
- An equation is a sentence in shorthand. Each letter stands for a real quantity.
- The = sign means both sides are the same amount, like a level balance scale.
- To solve, undo operations with their opposites — and always do it to both sides.
- Keep units attached to numbers. They follow the same rules and catch your mistakes.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.