Where did $1/2$ of this come from?

Question

Work done by an external force $F$ upon a particle displacing from point 1 to point 2 is defined as $$ W_{12} = \int_1^2 F \cdot dr \, .$$

Kinetic energy and work-energy theorem: According to Newton's second law, $F = m dv/dt$ and hence \begin{align} \mathbf{F} \cdot d\mathbf{r} & = m \frac{d\mathbf{v}}{dt} \cdot d\mathbf{r} = m \frac{d\mathbf{v}}{dt} \cdot \mathbf{v}dt \\ (\text{this line}) \qquad & = m \frac{d}{dt} \left[ \frac{1}{2} \mathbf{v}\cdot \mathbf{v} \right] dt \\ & = d \left[ \frac{1}{2} m v^2 \right] \end{align}

This is a lesson on work-energy theorem in classical mechanics.

Answer 1

If:

$$ f(v) = \tfrac12v^2$$

then:

$$\begin{align} \frac{df(v)}{dt} &= \frac{df(v)}{dv} \frac{dv}{dt} \\ &= \frac{2v}{2} \frac{dv}{dt} \\ &= v \frac{dv}{dt} \end{align}$$

$$ \therefore v \frac{dv}{dt} = \frac{d(\tfrac12v^2)}{dt}$$

It’s just calculus. It’s a neat trick that we use in vector calculus for some derivations.