# How are displacement, velocity and acceleration related?

Jul 29, 2015

The displacement ($x$), velocity ($v$) and acceleration ($a$) are related by a set of simple equations and are called the kinematic equations. I am writing them below.

#### Explanation:

Assuming that acceleration is uniform and the motion is rectilinear (we don't need to use vector notation then), the equations are,

v = v""_0 + at

v^2 = v""_0^2 + 2a(x-x""_0)

$x = x {\text{_0 + v}}_{0} t + \frac{1}{2} a {t}^{2}$

Where symbols have their usual meanings.

If the motion is on a plane or in space, we need to use vector notations or different equations for each of the coordinate axes.
And if acceleration is a function of time and is not constant, the equations get a little complicated. Two important results to find position vector and velocity vector are given below,

vecr(t) = vecr""_0 + int_0^t vecv(t)dt , where, vecr""_0 is the initial position vector.

vecv(t) = vecv""_0 + int_0^t veca(t)dt, where vecv""_0 is the initial velocity vector.

Here, $\vec{a} \left(t\right)$ denotes that the acceleration vector is not constant but is a function of time.