Let there be a given mass \(m\), acted by a net force \(F\). If the mass travels a small distance \(dr\), prove that the infinitesimal change in energy \(dE\) equals the work done by \(F\).

Note that \(m, E\) are scalars while \(F, dr\) are vectors.

**Solution**

We begin by defining \(F = \frac{dp}{dt}\) and \(dr = v dt\), where \(v\) is the velocity-vector.

Thus, \[Fdr = \frac{dp}{dt}\cdot v dt.\]

In relativistic mechanics, \(E = \sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}\)

hence,

\[\frac{dE}{dt} = \frac{p{c}^{2}}{\sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}} \cdot \frac{dp}{dt}\]

This long expression can be reduced to \(\frac{p{c}^{2}}{E} \cdot \frac{dp}{dt}\), which can be further reduced to \(v\cdot \frac{dp}{dt}\).

Assembling the above results yield \(\frac{dE}{dt} = v\cdot\frac{dp}{dt}\).

Therefore, \[dE = F\cdot dr.\]

Check out my other notes at Proof, Disproof, and Derivation

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TopNewestElegant!! Good job

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