# How does work relate to speed?

Oct 25, 2015

It could be through the Work and Kinetic Energy theorem:

#### Explanation:

Consider a constant force $\vec{F}$ acting on an object of mass $m$ for a distance $d$ along the $x$ axis only and producing an acceleration $a$;
work will be:

$W = F \cdot d = = m a \cdot d$ (from Newton's Law);

We can use cinematics to relate final and initial velocity to distance and accekeration as:

${v}_{f}^{2} = {v}_{i}^{2} + 2 a d$
And
$a d = \frac{{v}_{f}^{2} - {v}_{i}^{2}}{2}$
Substituting into the expression for work:
$W = m \frac{{v}_{f}^{2} - {v}_{i}^{2}}{2} = \frac{1}{2} m {v}_{f}^{2} - \frac{1}{2} m {v}_{i}^{2} = \Delta K$
Where $K$ is Kinetic Energy!