How to rearrange physics equations
By Mehmood Kakar · Updated
Quick answer
Do the same thing to both sides until the letter you want is on its own. Undo each operation with its opposite: × with ÷, + with −, and a square with a square root. For example, to find R in V = IR, divide both sides by I to get R = V ÷ I.
Why rearranging matters
Your exam equation sheet gives every equation in one fixed form. V = IR tells you how to find potential difference, but a question might give you V and I and ask for R. If you can't rearrange, you're stuck, even though the right equation is printed in front of you.
Rearranging is also where a lot of calculation marks go. Students pick the right equation, then divide when they should multiply, or forget a square root at the end. The method below works for every equation on the GCSE sheet.
The balance method
Think of an equation as a balance. Whatever you do to one side, you must do to the other, or it stops being true. To get a letter on its own, undo whatever is being done to it, using the opposite operation:
| If the letter is… | Undo it by… | Example |
|---|---|---|
| multiplied by something | dividing both sides by it | V = IR → R = V ÷ I |
| divided by something | multiplying both sides by it | ρ = m ÷ V → m = ρ × V |
| added to something | subtracting it from both sides | a + b = c → a = c − b |
| squared | square-rooting both sides | v² = 900 → v = 30 |
| multiplied by ½ | multiplying both sides by 2 | Ee = ½ke² → 2Ee = ke² |
When a letter has several things done to it, undo them in reverse order. In ½mv², the speed is squared first and then multiplied by ½ and m. So you undo the ½ and the m first, and take the square root last.
Five worked examples
1. Find resistance from V = IR
- V = I × R
- R is multiplied by I, so divide both sides by I
- R = V ÷ I
2. Find height from Ep = mgh
- Ep = m × g × h
- h is multiplied by m and by g, so divide both sides by both
- h = Ep ÷ (m × g)
The brackets matter: you divide by the product m × g, not by m and then multiply by g. Check this with the GPE calculator.
3. Find speed from Ek = ½mv²
- Ek = ½ × m × v²
- Multiply both sides by 2: 2Ek = m × v²
- Divide both sides by m: 2Ek ÷ m = v²
- Square-root both sides
- v = √(2Ek ÷ m)
This is the one students get wrong most often, usually by stopping at v² and forgetting the square root. The kinetic energy calculator shows each of these steps with your own numbers.
4. Find current from P = I²R (Higher)
- P = I² × R
- Divide both sides by R: P ÷ R = I²
- I = √(P ÷ R)
5. Find acceleration from v² − u² = 2as (Higher)
- v² − u² = 2 × a × s
- a is multiplied by 2 and by s, so divide both sides by 2s
- a = (v² − u²) ÷ 2s
To find the starting speed u instead, add u² to both sides, subtract 2as from both sides, then square-root: u = √(v² − 2as).
Formula triangles
A formula triangle puts the single quantity on top and the two multiplied quantities underneath. For s = vt, distance goes on top, with speed and time on the bottom. Cover the quantity you want and the triangle shows the calculation: cover v and you see s over t, so v = s ÷ t.
Triangles work for three-quantity equations such as V = IR, s = vt, Q = It and ρ = m ÷ V. They break down when an equation has a square, a ½, or an added term, which rules out kinetic energy, elastic potential energy and v² − u² = 2as. Use triangles if they help, but learn the balance method as well, because the harder equations are often worth more marks.
Rearrange or substitute first?
You can either rearrange the equation first and then put the numbers in, or put the numbers in first and then solve. Both can score full marks.
Substituting first is often easier if algebra worries you. For example, with Ep = mgh, 588 J, 60 kg and g = 9.8 N/kg: write 588 = 60 × 9.8 × h, work out 60 × 9.8 = 588, so 588 = 588 × h and h = 1 m. Whichever order you choose, write the substitution down clearly. A correct substitution can earn a mark even if the final answer goes wrong.
Check your answer
Three quick checks catch most rearranging mistakes:
- Put your answer back in. Substitute it into the original equation. If both sides don't match, something went wrong.
- Ask whether it's sensible. A person running at 300 m/s or a 2,000 kg tennis ball means you divided the wrong way or missed a square root.
- Check the unit. Rearranging V = IR for R should give ohms (Ω). If you are finding a time, the answer should be in seconds.
Practice questions
Rearrange each equation to make the given letter the subject. Click to see the answer.
1. Make I the subject of Q = It
I = Q ÷ t
2. Make m the subject of ρ = m ÷ V
m = ρ × V
3. Make k the subject of F = ke
k = F ÷ e
4. Make m the subject of Ek = ½mv²
m = 2Ek ÷ v²
5. Make e the subject of Ee = ½ke² (elastic potential energy)
e = √(2Ee ÷ k)
Frequently asked questions
Do I need to rearrange equations in GCSE Physics?
Yes. The equation sheet gives each equation in one form, but questions often ask for a different quantity, such as resistance from V = IR. You have to rearrange the equation, or substitute the numbers and solve, to get the answer.
Are formula triangles allowed in the exam?
Yes, you can use any correct method. Formula triangles only work for equations with three quantities and no squares, halves or added terms, so you still need algebra for equations like Ek = ½mv².
How do I rearrange an equation with a square?
Get the squared term on its own first, then take the square root of both sides. For Ek = ½mv², multiply by 2, divide by m, then square root: v = √(2Ek ÷ m).
Should I rearrange before or after substituting numbers?
Either can score full marks. Substituting first is easier if you find algebra hard; rearranging first means fewer numbers to carry through. Show the substitution clearly either way.