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If the kinetic energy of a body is directly proportional to time $t$, the magnitude of force acting on the body is
$(i)$ directly proportional to $\sqrt t$
$(ii)$ inversely proportional to $\sqrt t$
$(iii)$ directly proportional to the speed of the body
$(iv)$ inversely proportional to the speed of body
$(i), (ii)$
$(i), (iii)$
$(ii), (iv)$
$(i), (iv)$
Solution
$\mathrm{K} \propto \mathrm{t} \Rightarrow \mathrm{K}=\lambda \mathrm{t}, \lambda$ constant
$\frac{1}{2} \mathrm{mv}^{2}=\lambda \mathrm{t} \Rightarrow \mathrm{v}=\sqrt{\frac{2 \lambda}{\mathrm{m}}} {t}^{1 / 2}$
$\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}=\sqrt{\frac{2 \lambda}{\mathrm{m}}} \frac{1}{2} \mathrm{t}^{-1 / 2}$
$\mathrm{F}=\mathrm{ma}=\sqrt{\frac{\lambda \mathrm{m}}{2}} \frac{1}{\mathrm{t}^{1 / 2}} \Rightarrow \mathrm{F} \propto \frac{1}{\sqrt{\mathrm{t}}},$ $(ii)$ is $\mathrm{O} . \mathrm{K}$
$\mathrm{F}=\sqrt{\frac{\lambda \mathrm{m}}{2}} \frac{1}{\mathrm{t}^{1 / 2}}=\sqrt{\frac{\lambda \mathrm{m}}{2}} \times \frac{1}{\mathrm{v}} \sqrt{\frac{2 \lambda}{\mathrm{m}}}$
$=\Rightarrow \mathrm{F} \propto \frac{1}{\mathrm{v}},(\mathrm{iv}) \text { is } \mathrm{O} \cdot \mathrm{K}$
