A) \[{{x}^{2}}\]
B) \[{{e}^{x}}\]
C) \[x\]
D) \[{{\log }_{e}}x\]
Correct Answer: A
Solution :
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]
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