How to Rearrange a Physics Formula for Any Variable
Short answer: treat the formula like a recipe written for one variable and undo it step by step. Whatever was done to the unknown last, undo first — subtract what was added, divide what was multiplied, take the square root of a square — doing the same thing to both sides each time.
The method in four steps
- Circle the unknown. Find every place it appears.
- List what happens to it. In v = v₀ + a·t, the time t is multiplied by a, then v₀ is added.
- Undo in reverse order. Subtract v₀ from both sides: v − v₀ = a·t. Divide both sides by a: t = (v − v₀) / a.
- Check with units. (m/s − m/s) ÷ m/s² gives seconds, which is what time should be.
Squares and square roots
In K = ½·m·v², solving for v means: multiply both sides by 2, divide by m, then take the square root: v = √(2K / m). Remember that the square root has a positive and a negative answer; for a speed you keep the positive one.
Fractions and the unknown in the denominator
For P = F / A solved for A, multiply both sides by A and divide by P: A = F / P. A quick rule: if the unknown is on the bottom, swap it with the other side.
Common mistakes
- Dividing only one term: in v² = v₀² + 2·a·x you cannot divide just v₀² by 2a. Move v₀² first.
- Forgetting brackets: t = v − v₀ / a is not the same as t = (v − v₀) / a.
- Mixing units before rearranging: convert to SI units first or keep everything symbolic.
Checking your work on your phone
In Newtora you tap the variable you want to find. When it is not the subject of the formula, the solution shows a "Solve for" step with the rearranged equation before any numbers go in, so you can compare it with your own working.
Frequently asked questions
Should I rearrange before or after plugging in numbers?
Rearrange first. A symbolic answer like t = (v − v₀)/a is easy to check with units and you only do the arithmetic once.
What if the unknown appears twice?
Collect every term with the unknown on one side and factor it out. If it is squared in one term and not in another, you have a quadratic and may need the quadratic formula.
How do I know my rearranged formula is right?
Check the units of both sides, then plug in a quick test case where you already know the answer.