Page:LewisRevision.djvu/7

Its momentum and kinetic energy will change according to (11) and (12) by the amounts

$d\mathrm {M} =fdt$, $d\mathrm {E} '=fdl=fvdt$.

Hence

$d\mathrm {E} '=vd\mathrm {M}$.

(13)

So far the equations are those of Newtonian mechanics, but now in substituting for M from equation (10) we must regard m as a variable and write

$d\mathrm {E} '=mv\ dv+v^{2}dm$.

(14)

This will be our fundamental equation connecting the kinetic energy of a body with its mass and velocity.

Introducing now the relation of mass to energy given in equation (7) we may write,

$d\mathrm {E} '=\mathrm {V} ^{2}dm$,

and combining this equation with (14) gives

$\mathrm {V} ^{2}dm=mv\ dv+v^{2}dm$.

This equation, containing only two variables, m and v and the constant V, may readily be integrated as follows. By a simple transformation

$\left(1-{\frac {v^{2}}{\mathrm {V} ^{2}}}\right)dm={\frac {mvdv}{\mathrm {V} ^{2}}}$.

Writing β=v/V, and noting that

${\frac {vdv}{\mathrm {V} ^{2}}}=-{\frac {1}{2}}d\left(1-\beta ^{2}\right)$,

we see that

${\frac {dm}{m}}=-{\frac {1}{2}}{\frac {d\left(1-\beta ^{2}\right)}{\left(1-\beta ^{2}\right)}}$.

Hence

$\log m=-{\frac {1}{2}}\log \left(1-\beta ^{2}\right)+\log m_{0}$,

where log m₀ is the integration constant. Therefore

$\log {\frac {m}{m_{0}}}=\log \left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}$

or

${\frac {m}{m_{0}}}={\frac {1}{\left(1-\beta ^{2}\right)^{1/2}}}$.

(15)

This is the general expression for the mass of a moving body in terms of β, the ratio of its velocity to the velocity of