A) \[{{x}^{2}}\]
B) \[{{e}^{x}}\]
C) \[x\]
D) \[lo{{g}_{e}}x\]
Correct Answer: A
Solution :
From given information\[a=-kx\]where a is acceleration,\[x\]is displacement, and k is a proportionality constant. \[\frac{v\,dv}{dx}=-kx\] \[\Rightarrow \] \[v\text{ }dv=-k\text{ }x\text{ }dx\] Let for any displacement from 0 to\[x,\]the velocity changes from\[{{v}_{0}}\]to\[v\]. \[\Rightarrow \] \[\int_{{{v}_{0}}}^{v}{vdv}=-\int_{0}^{x}{k\,x\,dx}\] \[\Rightarrow \] \[\frac{{{v}^{2}}-v_{0}^{2}}{2}=-\frac{k{{x}^{2}}}{2}\] \[\Rightarrow \] \[m\left( \frac{{{v}^{2}}-v_{0}^{2}}{2} \right)=-\frac{mk{{x}^{2}}}{2}\] \[\Rightarrow \] \[\Delta K\propto {{x}^{2}}\] [\[\Delta K\]is loss in KE]
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