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

The velocity-displacement graph of a particle is shown in the figure. Give the relation between v and x, and also Obtain the relation between acceleration and displacement and plot it.

Answers (1)

a) Consider the point P(x,v) at any time t on the graph such that angle ABO is \theta such that

\tan \theta = \frac{AQ}{QP} = \frac{(v_{0}-v)}{x} = \frac{v_{0}}{x_{0}}

When the velocity decreases from v_{0}  to zero during the displacement, the acceleration becomes negative.

v_{0}-v=\left ( \frac{v_{0}}{x_{0}} \right )x

So, the relation between v and x is given by,

v=v_{0}(1-\frac{x}{x_{0}})

Now, to obtain the relation between acceleration and displacement,

a = \frac{dv}{dt} = (\frac{dv}{dt})(\frac{dx}{dx}) = \left ( \frac{dv}{dx} \right )\left ( \frac{dx}{dt} \right )

a=\frac{-v_{0}}{x_{0}}v

a=\left ( \frac{v{_{0}}^{2}}{x{_{0}}^{2}} \right )x-\left ( \frac{v{_{0}}^{2}}{x_{0}}\right )

At \; x = 0

a=\frac{-v{_{0}}^{2}}{x^{0}}

At \; a = 0

x=x_o

The points are,

(0, \frac{-v{_{0}}^{2}}{x_{0}})and B(x_{0},0)

Posted by

infoexpert22

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