A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

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TestHub · Accepted Answer

Consider a small steel ball as shown below:

As shown in diagram, buoyancy force $F_{B} = F$ .

Applying Newton's second law, we get

$m g - F_{B} - F_{v} = m a$

$\Rightarrow (ρ \frac{4}{3} π r^{3}) g - (ρ_{L} \frac{4}{3} π r^{3}) g - 6 π η r v = m \frac{d v}{d t}$

Let $\frac{4}{3 m} π R^{3} g (ρ - ρ_{L}) = K_{1}$ and $\frac{6 π η r}{m} = K_{2}$ , then we get

$\Rightarrow \frac{d v}{d t} = K_{1} - K_{2} v$

Integrating both sides, we get

$\Rightarrow \int_{0}^{v} \frac{d v}{K_{1} - K_{2} v} = \int_{0}^{t} d t$

$\Rightarrow - \frac{1}{K_{2}} \ln {[K_{1} - K_{2} v]}_{0}^{v} = t$

$\Rightarrow \ln (\frac{K_{1} - K_{2} v}{K_{1}}) = - K_{2} t$

$\Rightarrow K_{1} - K_{2} v = K_{1} e^{- K_{2} t}$

$\Rightarrow v = \frac{K_{1}}{K_{2}} [1 - e^{- K_{2} t}]$

Clearly, option B matches with the given variation of velocity with time.

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