In an experiment to verify Stokes law, a small spherical ball of radius r and density $ρ$ falls under gravity through a distance h in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of h is proportional to : (ignore viscosity of air)

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$r^{4}$

$r^{2}$

$r^{3}$

$r$

answer is A.

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Detailed Solution

Velocity of the ball after free fall : $V = \sqrt{2 g h}$
given it is equivalent to terminal velocity in a liquid
$\begin{array}{l} m g - B = F_{d r a g} \\ \Rightarrow ρ \frac{4}{3} π r^{3} g - J \frac{4}{3} π r^{3} g = 6 π η r V_{T} \\ \Rightarrow (ρ - J) \frac{4}{3} π r^{2} g = 6 π η \sqrt{2 g h} \\ \Rightarrow h \propto r^{4} \end{array}$

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