A solid metallic sphere of radius $r$ is allowed to fall freely through air. If the frictional resistance due to air is proportional to the cross-sectional area and to the square of the velocity, then the terminal velocity of the sphere is proportional to which of the following?
Diffcult
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The frictional resistance $f_{r}$ is given as$:$
$f_{r} \propto A$ and $f_{r} \propto v^{2}$
$f_{r}=k A v^{2}(\text { here } \mathrm{k}$ is a constant)
The downwards force is given as$:$ $m g=\frac{4}{3} \pi r^{3} \rho g$
Balancing the above we get the terminal velocity as$:$
$\frac{4}{3} \pi r^{3} \rho g=k\left(\pi r^{2}\right) v^{2}$
$r \propto v^{2}$
$v \propto r^{1 / 2}$
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