The velocity- displacement graph of a particle is shown in 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 figure.
$(a)$ Write the relation between $v$ and $x$.
$(b)$ Obtain the relation between acceleration and displacement and plot it.
From graph, initial velocity $=v_{0}$ and distance travelled in time $t=x_{0}$.
For the graph $\tan \theta=\frac{v_{0}}{x_{0}}=\frac{v_{0}-v}{x} \quad \ldots$ (1)
Where, $v$ is velocity and $x$ is displacement at any instant of time $t$.
From Eq. (i) $\quad v_{0}-v=\frac{v_{0}}{x_{0}} x$ $\Rightarrow \quad v=\frac{-v_{0}}{x_{0}} x+v_{0} \ldots$
$(2)$ We know that
Acceleration $a=\frac{d v}{d t}=\frac{-v_{0}}{x_{0}} \frac{d x}{d t}+0$
$a=\frac{-v_{0}}{-}(v)$
$=\frac{x_{0}}{x_{0}}\left(\frac{-v_{0}}{x_{0}} x+v_{0}\right)$
$=\frac{v_{0}^{2}}{x_{0}^{2}} x-\frac{v_{0}^{2}}{x_{0}}$
$(\because$ From $(2))$
Graph of a versus $x$ is given above.
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