Question

How are displacement, velocity and acceleration related?

Answer 1

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""_0 + v""_0t + 1/2at^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, `veca(t)` denotes that the acceleration vector is not constant but is a function of time.