# What is the formula for distance from a changing velocity?

Oct 6, 2014

If the velocity is changing at a constant rate, meaning that the acceleration is constant, we may derive the formula as follows.

The velocity $v$ varies linearly with time and is given by the relation $v \left(t\right) = {v}_{0} + a t$ where ${v}_{0}$ is the initial velocity and $a$ is the acceleration.

We know that distance is the product of average velocity and time, and the average velocity is the average of the initial and final velocity. In mathematical terms,

${v}_{a v g} = \setminus \frac{{v}_{0} + v}{2} = \setminus \frac{{v}_{0} + \left({v}_{0} + a t\right)}{2} = {v}_{0} + \setminus \frac{a t}{2}$

Simply multiply that by time $t$ to get the distance $s$.

$s = \left({v}_{0} + \frac{a t}{2}\right) \cdot t = {v}_{0} t + \frac{1}{2} a {t}^{2}$