• # question_answer 1) A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement  $x$ is proportional to A) ${{x}^{2}}$ B)  ${{e}^{x}}$ C)  $x$ D)  ${{\log }_{e}}x$

From given information $a=-\,kx,$where a is acceleration, $x$ is displacement and $k$ is a proportionality constant. $\frac{v\,dv}{dx}=-\,k\,x$ $\Rightarrow$ $v\,dv=-\,k\,x\,dx$ Let for any displacement from 0 to $x,$ the velocity changes from ${{v}_{0}}$to v. $\Rightarrow$ $\int_{{{v}_{0}}}^{v}{v\,dv=-\int_{0}^{x}{k\,x\,dx}}$ $\Rightarrow$ $\frac{{{v}^{2}}-v_{0}^{2}}{2}=-\frac{k\,{{x}^{2}}}{2}$ $\Rightarrow$ $m\left( \frac{{{v}^{2}}-v_{0}^{2}}{2} \right)=-\frac{mk\,{{x}^{2}}}{2}$ $\Rightarrow$ $\Delta K\propto {{x}^{2}}$ [$\Delta K$is loss in KE]