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# Relativistic Force

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

Note by Steven Zheng
2 years, 10 months ago

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