Relativistic Force

Let there be a given mass mm, acted by a net force FF. If the mass travels a small distance drdr, prove that the infinitesimal change in energy dEdE equals the work done by FF.

Note that m,Em, E are scalars while F,drF, dr are vectors.

Solution

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

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

In relativistic mechanics, E=(pc)2+(mc2)2E = \sqrt{{(pc)}^{2} + {(m{c}^{2})}^{2}}

hence,

dEdt=pc2(pc)2+(mc2)2dpdt\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 pc2Edpdt\frac{p{c}^{2}}{E} \cdot \frac{dp}{dt}, which can be further reduced to vdpdtv\cdot \frac{dp}{dt}.

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

Therefore, dE=Fdr.dE = F\cdot dr.

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

Note by Steven Zheng
4 years, 11 months ago

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Elegant!! Good job

Racchit Jain - 3 years, 5 months ago

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