Question

Prove Work Energy Theorem by a variable force.

Answer 1

$\Delta W=F\left(x\right)\Delta x$

$W={\sum }_{x_i}^{x_f}F\left(x\right)\Delta x$

$W={lim}_{\Delta x\to 0}{\sum }_{x_i}^{x_f}F\left(x\right)\Delta x$

or,

$W={\int }_{x_i}^{x_f}F\left(x\right)dx$

$K=\frac{1}{2}m{v}^2$

$\frac{dK}{dt}=m\frac{dv}{dt}\cdot v$

$\Rightarrow \frac{dK}{dt}=F\frac{dx}{dt}$

$\Rightarrow \Delta K={\int }_{x_i}^{x_f}F\cdot dx$