Graph of velocity versus displacement of a particle moving in a straight line is as shown in figure.

The acceleration of the particle is

 

From the graph, velocity-displacement equation can be written as:

$\mathrm{v} = {\mathrm{v}}_{\mathrm{o}}+\mathrm{\alpha x} ....\left(1\right)$

$\mathrm{Here} {\mathrm{v}}_{\mathrm{o}} \mathrm{and} \mathrm{\alpha } \mathrm{are} \mathrm{positive} \mathrm{constants}.$

Differentiating (1) with respect ot x we get

$\frac{\mathrm{dv}}{\mathrm{dx}} = \mathrm{\alpha } = \mathrm{constant}$

Acceleration of the particle can be written as

$\mathrm{a} = \mathrm{v}.\frac{\mathrm{dv}}{\mathrm{dx}} = ({\mathrm{v}}_{\mathrm{o}}+\mathrm{\alpha x}).\mathrm{\alpha }$

$\mathrm{\alpha }-\mathrm{x}$ equation is a linear equation. Thus acceleration increases linearly with x.