Question
State work-energy theorem.
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Answer
Work energy theorem states that the change in kinetic energy of an object is equal to the net work done on it by the net force. Let us suppose that a body is initially at rest and a forceis applied on the body to displace it through along the direction of the force. Then, small amount of work done is given by $\text{dW}=\overrightarrow{\text{F}}\times\overrightarrow{\text{dS}}=\text{FdS}$ Also, according to Newton's second law of motion, we have $F =$ ma where a is acceleration produced (in the direction of force) on applying the force. Therefore, $\text{dW}=\text{MadS}=\text{m}\frac{\text{dv}}{\text{dt}}\text{ds}$ Or $\text{dW}=\text{m}\frac{\text{ds}}{\text{dt}}\text{dv}=\text{Mvdv}$ Now, work done by the force in order to increase its velocity from $u ($initial velocity$)$ to $v$ (final velocity) is given by $\text{W}=\int\limits^\text{v}_\text{u}\text{Mvdv}=\text{M}\int^\text{v}_\text{u}\text{vdv}$ $=\text{M}\big|\frac{\text{v}^2}{2}\big|^\text{v}_\text{u}$ $\text{W}=\frac{1}{2}\text{Mv}^2-\frac{1}{2}\text{Mu}^2$ Hence, work done on a body by a force is equal to the change in its kinetic energy.
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