動能定理者,物理定理也。其述乎功可以更動能以等量。
其式曰 W=ΔEk=Ek−Ek0=12mv2−12mu2{\displaystyle W=\Delta E_{k}=E_{k}-E_{k0}={\frac {1}{2}}mv^{2}-{\frac {1}{2}}mu^{2}},或書W=∫F→ds→=12mv2−12mu2{\displaystyle W=\int {\vec {F}}\mathrm {d} {\vec {s}}={\frac {1}{2}}mv^{2}-{\frac {1}{2}}mu^{2}}。
W{\displaystyle W}
=∫F→ds→{\displaystyle =\int {\vec {F}}\mathrm {d} {\vec {s}}}
=∫ma→ds→{\displaystyle =\int m{\vec {a}}\mathrm {d} {\vec {s}}}
=∫mdv→dtds→{\displaystyle =\int m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}\mathrm {d} {\vec {s}}}
=∫mds→dtdv→{\displaystyle =\int m{\frac {\mathrm {d} {\vec {s}}}{\mathrm {d} t}}\mathrm {d} {\vec {v}}}
=∫mv→dv→{\displaystyle =\int m{\vec {v}}\mathrm {d} {\vec {v}}}
=12mv2−12mu2{\displaystyle ={\frac {1}{2}}mv^{2}-{\frac {1}{2}}mu^{2}}