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W = ∫ F d r = ∫ m a d r = ∫ m d v d t d r = m ∫ d r d t d v = m ∫ v d v = m v 2 2 = Δ E k {\displaystyle {\begin{aligned}W=\int F\,dr&=\int ma\,dr\\&=\int m\,{\frac {dv}{dt}}\,dr\\&=m\int {\frac {dr}{dt}}\,dv\\&=m\int v\,dv\\&={\frac {mv^{2}}{2}}=\Delta E_{k}\end{aligned}}}
