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A small ball of mass ‘m’& radius r is released from rest in a viscous liquid of coefficient of viscosity $η$ and density half of that of ball. Choose the correct option(s)

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The ball will never attain constant velocity

After time $t = \frac{m l n 2}{6 π η r}$ the velocity of ball will be half of the terminal velocity

The instantaneous velocity of ball at a time t is given by $v = v_{T} (1 - e^{\frac{6 π η r}{m} t})$

The velocity of ball will remain constant

answer is B.

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Detailed Solution

$m a = m g - \frac{M G}{2} - 6 π η r v$

$m a = 6 π η r v_{r} - 6 π η r$

$\frac{dv}{dt} = \frac{6πηr}{m} (V_{T} -V)$

$\frac{dv}{V_{T} -V} = \frac{6πηr}{m} d t$

$\frac{V_{T} -V}{V_{T}} =e \frac{6πηr}{m} t$

$V = V_{T} (1 - e^{\frac{6 π η r}{m} t})$

$\frac{1}{2} = 1 - e^{- \frac{6 π η r}{m} t}$

$\frac{6 π η r t}{m} = l n 2$

$t = \frac{m l n 2}{6 π η r}$

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