Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 14 / a / i

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Solution

Take the total differential of the logarithm of the relativistic energy, with the speed of light fixed:

ln E = ln m + 2 ln c - \frac{1}{2} ln (1 - \frac{v ^{2}}{c ^{2}}) .

(1)

Using the chain rule,

\frac{d E}{E} = \frac{d m}{m} + \frac{v d v}{c ^{2} - v ^{2}} = \frac{d m}{m} + \frac{v ^{2}}{c ^{2} - v ^{2}} \frac{d v}{v} .

(2)

Every term is a dimensionless fractional change. The differential relation is exact; applying it to finite small changes is a first-order Taylor expansion.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge
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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 14 / a / i

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