# How to prove equation of kinetic energy (KE = 1/2mv^2)?

Jan 20, 2018

See below.

#### Explanation:

The work done is accelerating an object is given by:

$W = F \Delta x$

Where $F$ is the force and $\Delta x$ the displacement.

If the object started from rest and all of the work was converted to kinetic energy then this will be equal to the kinetic energy of the object:

$K = F \Delta x$

Using Newton's 2nd law:

$K = m a \Delta x = m \left(a \Delta x\right)$

Now using the equation of motion:

$2 a \Delta x = {v}^{2} - {v}_{0}^{2} \to a \Delta x = {v}^{2} / 2 - {v}_{0}^{2} / 2$

Substitute this into the equation for kinetic energy to get:

$K = m \left({v}^{2} / 2 - {v}_{0}^{2} / 2\right)$

If the object started from rest then the initial velocity will be:

${v}_{0} = 0$ so $K$ simplifies to:

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