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?
JEE MAIN 2024, Diffcult
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$\mathrm{mg}-\mathrm{F}_{\mathrm{B}}-\mathrm{F}_{\mathrm{v}}=\mathrm{ma}$
$\left(\rho \frac{4}{3} \pi \mathrm{r}^3\right) \mathrm{g}-\left(\rho_{\mathrm{L}} \frac{4}{3} \pi \mathrm{r}^3\right) \mathrm{g}-6 \pi \eta \mathrm{r}=\mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}}$
$\text { Let } \frac{4}{3 \mathrm{~m}} \pi \mathrm{R}^3 \mathrm{~g}\left(\rho-\rho_{\mathrm{L}}\right)=\mathrm{K}_1 \text { and } \frac{6 \pi \eta \mathrm{r}}{\mathrm{m}}=\mathrm{K}_2$
$\frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{K}_1-\mathrm{K}_2 \mathrm{v}$
$\int_0^{\mathrm{v}} \frac{\mathrm{dv}}{\mathrm{K}_1-\mathrm{K}_2 \mathrm{v}}=\int_0^{\mathrm{t}} \mathrm{dt}$
$-\frac{1}{\mathrm{~K}_2} \ln \left[\mathrm{K}_1-\mathrm{K}_2 \mathrm{v}\right]_0^{\mathrm{v}}=\mathrm{t}$
$\ln \left(\frac{\mathrm{K}_1-\mathrm{K}_2 \mathrm{v}}{\mathrm{K}_1}\right)=-\mathrm{K}_2 \mathrm{t}$
$\mathrm{K}_1-\mathrm{K}_2 \mathrm{v}=\mathrm{K}_1 \mathrm{e}^{-\mathrm{K}_2 \mathrm{t}} $
$\mathrm{v}=\frac{\mathrm{K}_1}{\mathrm{~K}_2}\left[1-\mathrm{e}^{-\mathrm{K}_2 \mathrm{t}}\right]$
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