Every term in a correct equation has the same units. If you write the units of each side and they do not match, the formula is wrong, and no amount of careful arithmetic will rescue it.
This lesson is part of selecting and checking a physical model in SPM Physics. It works alongside the units and significant figure checker.
What is the two-line check?
Use two lines.
- Write the units of each quantity in the formula, using base units: m, kg and s.
- Combine them on each side and compare.
Start with s = ut + ½at². The left side is in m. The term ut has units (m s⁻¹)(s) = m, and the term ½at² has (m s⁻²)(s²) = m. Every term is in metres, so the check passes.
Worked example 1: a misremembered formula
A student writes s = ut + ½at, with the square missing from t. The term ½at has units (m s⁻²)(s) = m s⁻¹.
The equation adds a length (ut) to a velocity (½at), which is not allowed. The check fails, so the formula is wrong before any numbers are used. The fix is to restore t².
Worked example 2: a wrong rearrangement
A student knows E = ½mv² and needs v for E = 200 J and m = 4.0 kg. They write v = √(2Em) and get √(2 × 200 × 4.0) = √1600 = 40.
The units of 2Em are J kg = kg² m² s⁻². Its square root is kg m s⁻¹, which is a momentum, not a speed. The check fails.
The correct rearrangement is v = √(2E ÷ m). Substitute: √(400 ÷ 4.0) = √100 = 10 m s⁻¹. The units of 2E ÷ m are (kg m² s⁻²) ÷ kg = m² s⁻², and the square root is m s⁻¹.
Worked example 3: impulse
Impulse is Ft, with units N s = kg m s⁻². Multiplying by s gives kg m s⁻¹, the same as momentum mv. That is why impulse equals the change in momentum.
If you wrote impulse = Fs instead, the units would be N m, which is energy. The check shows that F × distance is work, not impulse.
The mistake to avoid
The common mistake is to trust the check when it passes and ignore numerical factors. The table shows what the check can and cannot find.
| Error | Found by the unit check? |
|---|---|
| Missing t² | Yes |
| Wrong rearrangement (Em instead of E ÷ m) | Yes |
| Missing factor ½ | No |
| Using km h⁻¹ instead of m s⁻¹ | Only if the units are written out |
After a passing check, test the formula on a simple case, such as a = 0, where s should equal ut.
Check yourself
A student says the force needed to stop a trolley is F = mv × t. Use units to decide whether this is plausible.
Answer
Units of mv × t = (kg m s⁻¹)(s) = kg m, which is not a force. A force has units kg m s⁻² = N.
The correct relation is Ft = mv − mu, so F = (mv − mu) ÷ t. The units of (kg m s⁻¹) ÷ s = kg m s⁻², which is N.
What to study next
Continue with rejecting a mathematically valid answer that violates the stated physical setup. Record the formula errors you catch in the mistake log and paper-error review tool.
If you want a teacher to go through your unit checks with you, see online one-to-one Physics tuition.