question_answer 40)

                  A body of mass 0.5 kg travels in a straight line with velocity \[v=a\,{{x}^{3/2}}\] where \[a=5\,\,{{m}^{-1/2}}\,{{s}^{-1}}\]. The work done by the net force during its displacement from \[x=0\] to \[x=2\,m\] is                 (a) 1.5 J                                 (b) 50 J                 (c) 10 J                                  (d) 100 J                

Answer:

                  (b) \[W=\,\int_{{}}^{{}}{F\,dx}\,\,=\int_{{}}^{{}}{m\frac{dy}{dt}\,.\,dx}\]                                 \[=\,\int_{{}}^{{}}{mv\,\,dv}\,\]                        \[\,\left( \because \frac{dx}{dt}=\,\nu  \right)\]                                 \[\nu =\,a{{x}^{3/2}}\,=\,5{{x}^{3/2}}\]                 When    \[x=0,\,v=0\]                 When \[x=2,\,v=10\sqrt{2}\,m\,{{s}^{-1}}\]                 \[\therefore \] \[W=\,m\,\int\limits_{0}^{10\sqrt{2}}{ndv}\,=\,0.5\,\left[ \frac{{{v}^{2}}}{2} \right]_{0}^{10\sqrt{2}}\]                 \[=\,\frac{0.5}{2}\times 100\,\times \,2\,=\,50\,J\]