Question

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)?

Answer 1

Applying the equation, `v^2=u^2 +2as` (for constant acceleration `a`)

If the body started from rest,then,`u=0`,so total displacement, `s=v^2/(2a)` (where,`v` is the velocity after displacement `s`)

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

so,work done by force `F` in causing `d x` amount of displacement is `dW =F*dx`

so, `dW =madx`

or, `int_0^WdW =maint_0^s dx`

so,`W=ma[x]_0^(v^2/(2a))` (as, `s=v^2/(2a)`)

so,`W=ma(v^2)/(2a)=1/2mv^2`

Proved