# How can we prove that the work done to accelerate a body from rest to a velocity, V is given by W=1/2(mV^2)?

Mar 14, 2018

Applying the equation, ${v}^{2} = {u}^{2} + 2 a s$ (for constant acceleration $a$)

If the body started from rest,then,$u = 0$,so total displacement, $s = {v}^{2} / \left(2 a\right)$ (where,$v$ is the velocity after displacement $s$)

Now,if force $F$ acted on it,then $F = m a$ ($m$ is its mass)

so,work done by force $F$ in causing $d x$ amount of displacement is $\mathrm{dW} = F \cdot \mathrm{dx}$

so, $\mathrm{dW} = m a \mathrm{dx}$

or, ${\int}_{0}^{W} \mathrm{dW} = m a {\int}_{0}^{s} \mathrm{dx}$

so,$W = m a {\left[x\right]}_{0}^{{v}^{2} / \left(2 a\right)}$ (as, $s = {v}^{2} / \left(2 a\right)$)

so,$W = m a \frac{{v}^{2}}{2 a} = \frac{1}{2} m {v}^{2}$

Proved