Letters behave like numbers
A physics formula is written to answer one question. answers “what force do I need?”. But problems don't always ask that. Sometimes you know the force and the acceleration and want the mass. Then you rearrangeRewriting a formula so that a different letter is alone on one side. Also called 'solving for' or 'making the subject'. the formula so stands alone.
There's nothing new to learn: it's the balance rule from Equations Are Balances. Whatever you do to one side, do to the other. The only change is that the other letters stay as letters. Treat them exactly as if they were numbers you haven't been told yet.
Undo in reverse order
When the letter you want has several things done to it, peel them off in the opposite order to how they were applied, like getting undressed: shoes come off before socks. In , the letter was first multiplied by , then had added. So undo the adding first, then the multiplying:
Worked example: solve v = u + at for a
Step 1 of 4- 1Start. a is multiplied by t, and then u is added on.
Try it on four real formulas
Pick a formula and a letter. The steps show each undo move; the sliders let you plug in numbers and check that the result makes sense.
Choose a physics formula and the letter to solve for. The steps that get that letter on its own are listed, each one undoing an operation on both sides. Sliders set the other quantities, and the result is computed from the rearranged formula and checked against the original.
Solve for any letter
Interactive- Newton's second law: start from
- 1m is multiplied by a.
- 2Undo '× a': divide both sides by a.
- 3The a's cancel on the right. Swap sides to read it.
Formula
Solve for
The math, with your numbers
Put it back into the original formula: both sides come out equal.
Letters on the bottom, and squares
Two situations trip people up. First, when the letter you want is in the denominator, as is in : multiply it out of the bottom first, then deal with the rest.
Second, when the letter is squared, as is in : clear everything else away until is alone, and take the square root last.
Make a prediction first
Using v = √(2E/m): a 10 kg cart has 500 J of kinetic energy. How fast is it going?
Think about it, then click to reveal the answer.
Make a prediction first
Using v = √(2E/m): a 10 kg cart has 500 J of kinetic energy. How fast is it going?
Think about it, then click to reveal the answer.
. Check it the other way: ✓
Two free checks
Every rearranged formula can be checked in seconds, two ways:
- Put numbers back in. Choose values that satisfy the original (say ) and see whether your new formula gives the right letter back.
- Check the units. In , newtons divided by m/s² should give kilograms, and it does, because a newton is a kg·m/s². If the units come out as something else, a step went wrong.
Wrap-up
Key ideas to remember
- Rearranging is the balance rule with letters: do the same thing to both sides.
- Undo operations in the reverse order they were applied: last one done, first one undone.
- A letter on the bottom: multiply it out first. A squared letter: get the square alone, then take the root.
- Check by putting numbers back in, and by checking the units.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.