Ncert Exercises & Solutions

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.

Hint: Appy acceleration, a =

v \frac{dv}{dx} .

$v \frac{dv}{dx} .$

Step 1: Find the relation between v and x.

Given, initial velocity $= v_{0}$ $= v_{0}$
$For the graph tanθ = \frac{v_{0}}{x_{0}} = \frac{v_{0} - v}{x}$ $For the graph tanθ = \frac{v_{0}}{x_{0}} = \frac{v_{0} - v}{x}$ ...(I)
where, v is velocity and x is the displacement at any instant of time t.

Step 2: Find velocity as a function of displacement using the above relation.

$\begin{array}{lr} From Eq. (i) & v_{0} - v = \frac{v_{0}}{x_{0}} x \\ \Rightarrow & v = \frac{- v_{0}}{x_{0}} x + v_{0} \end{array}$

Step 3: Find acceleration by using the relation acceleration, $a = v \frac{dv}{dx} .$

$\begin{array}{l} Acceleration a = v \frac{dv}{dx} = (\frac{- v_{0}}{x_{0}} x + v_{0}) \times (\frac{- v_{0}}{x_{0}}) \\ = \frac{- v_{0}}{x_{0}} (\frac{- v_{0}}{x_{0}} x + v_{0}) \\ = \frac{v_{0}^{2}}{x_{0}^{2}} x - \frac{v_{0}^{2}}{x_{0}} . \end{array}$

Step 4: Plot the graph of a vs x.

The graph of a versus x is

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