Dimensional Reasoning Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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A student writes

E = mc

for Einstein's mass-energy formula. Use dimensional reasoning to explain why this cannot be correct and what the correct formula should be.

1
$[E]$ = energy = $\text{kg}\cdot\text{m}^2/\text{s}^2 = \text{J}$ .
2
$[m]$ = mass = kg. $[c]$ = speed of light = m/s.
3
$[mc] = \text{kg}\cdot\text{m}/\text{s}$ — units of momentum, not energy. Dimensionally wrong.
4
For dimensional consistency, we need $[mc^2] = \text{kg}\cdot(\text{m}/\text{s})^2 = \text{kg}\cdot\text{m}^2/\text{s}^2 = \text{J}$ . So the correct formula is $E = mc^2$ .

Answer

E = mc^2 \text{ (not } mc\text{; dimensional analysis reveals the error)}

Dimensional reasoning is a powerful error-detection tool. The incorrect formula

E = mc

has the wrong units (momentum, not energy), which immediately signals an error even without knowing the physics.

About Dimensional Reasoning

Using the units and dimensions of physical quantities to check formulas, guide derivations, and eliminate impossible answers.

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