The velocity-displacement graph of a particle is shown in Fig . (a) Write the relation between (v) and (x).
(b) Obtain the relation between acceleration and displacement and plot it .
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Explanation

Here `v_(0)` is initial velocity, if x and v be distance and velocity at any time t at A then from figure, slope is given by
`tan theta=-(v_(0))/(x_(0))=(v_(0)-v)/(x-0)`
or `v_(0)-v=(v_(0))/(x_(0))x`
or `v=v_(0)-(v_(0))/(x_(0))x` . . . . . . (i)

and acceleration `a=(dv)/(dt)=-(v_(0))/(x_(0))(dx)/(dt)=-(v_(0))/(x_(0))(v)`
`=-(v_(0)).(x_(0))(v_(0)-(v_(0))/(x_(0))x)`
Graph between a and x is shown in figure below.

or `a=-(v_(0)^(2))/(x_(0))+(v_(0)^(2))/(x_(0)^(2))x`