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12. If the unit of kinetic energy be that of a train of mass $m$ tons moving with a velocity of $v$ miles an hour, the unit of power that of an engine of horse-power $h$, and the unit of force the weight of $n$ tons, prove that the unit of mass is $ \frac12m\left(\frac{448vn}{75h}\right)^2$ tons.

Solution:

$\;\;\;$ With the usual notation we have
$$ [M'][L']^2[T']^{-2} = \frac12 \cdot 2240m \cdot \left(\frac{22}{15}v\right)^2 [M][L]^2[T]^{-2},$$
$$ [M'][L']^2[T']^{-3} = 32 \cdot 550h \cdot [M][L]^2[T]^{-3},$$
$$ [M'][L']^2[T']^{-2} = 32 \cdot 2240n \cdot [M][L]^2[T]^{-2},$$

Multiplying the first equation by the square of the third, and dividing by the square of the second, we get the required value for $[M']$.