A student writes the kinetic energy formula as E = ½ m vⁿ but forgets the exponent. Using SI base units, determine the value of n that makes the equation dimensionally correct.

Measurements Challenge #2 — Dimensional Analysis

Spot the
dimension error.

Measurements Medium

Problem

A student remembers the kinetic energy formula as \( E = \tfrac{1}{2} m v^{\,n} \) — but has forgotten the exponent. Using SI base units, determine the value of \( n \) that makes the equation dimensionally correct.

Energy unit\( \mathrm{J} = \mathrm{kg \, m^2/s^2} \)
Mass unit\( \mathrm{kg} \)
Speed unit\( \mathrm{m/s} \)
Answer asthe exponent n

Pure numbers like ½ carry no dimensions — ignore them.

Hint: write both sides in base units. The left side is \( \mathrm{kg\,m^2/s^2} \); the right is \( \mathrm{kg} \cdot (\mathrm{m/s})^n \). What n makes the metres and seconds match?

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Spot thedimension error.

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Spot the
dimension error.