Prove that change in kinetic energy of a paticle is equal to the work done on it by a variable force.

Explanation

Let 'K' represent kinetic energy. We know that `K=1//2 mv^2`, the time rate of change of kinetic energy is given by `(dK)/(dt)=(d)/(dt)((1)/(2)mv^2)=(1)/(2)m2v((dv)/(dt))=mv(dv)/(dt)=v(ma)=fv`
i.e., `(dK)/(dt)=F(dx)/(dt)` where `v=(dx)/(dt)=` time rate of displacement of particle.
or `dK=Fdx`
Integrating from the initial to final position,
we write `underset(k_i)overset(k_f)(int)=underset(x_i)overset(x_f)intFdx`
i.e., `(K_f-K_i)=underset(x_i)overset(x_f)int F(x)dx`
since `'Delta K'` is a scalar quantity, the direction contained in Newton's II law is absent.