question_answer 50)

                  The velocity ? displacement graph of a particle is shown in Fig.                 (a) Write the relation between \[\upsilon \]  and \[x\]                 (b) Obtain the relation between acceleration and displacement and plot it.                

Answer:

                  (a) Equation of straight line is given by                 \[y=mx+c\]                 Where, m is the slop and c is the intercept on y axis.                 In the given graph, \[y=\upsilon ,\]slope \[=\left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)\]                 And intercept \[={{\upsilon }_{0}}\]                 \ \[\upsilon \,=\,\left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)\,x\,+\,{{\upsilon }_{0}}\]                           ? (i)                 (b) \[a\,=\frac{d\upsilon }{dt}\,\,=\,\,\left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)\frac{dx}{dt\,}\,=\left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)\upsilon \]                 Using eqn. (i), we get                 \[a\,=\,\left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)\left[ \left( -\frac{{{\upsilon }_{0}}}{{{x}_{0}}} \right)x+{{\upsilon }_{0}} \right]\]                 \[\,=\,\left( \frac{\upsilon _{0}^{2}}{x_{0}^{2}} \right)x\,-\,\upsilon _{0}^{2}\,\]                 Graph between \['a'\]  and \['x'\]  is shown in figure