Motion In One Dimension Question 12
Question 12 - 2024 (31 Jan Shift 1)
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?
(1)
(2)
(3)
(4)
Show Answer
Answer: (2)
Solution:
$\mathrm{mg}-\mathrm{F}{\mathrm{B}}-\mathrm{F}{\mathrm{v}}=\mathrm{ma}$
$\left(\rho \frac{4}{3} \pi r^{3}\right) g-\left(\rho_{L} \frac{4}{3} \pi r^{3}\right) g-6 \pi \eta r v=m \frac{d v}{d t}$
Let $\frac{4}{3 \mathrm{~m}} \pi \mathrm{R}^{3} \mathrm{~g}\left(\rho-\rho_{\mathrm{L}}\right)=\mathrm{K}{1}$ and $\frac{6 \pi \eta \mathrm{r}}{\mathrm{m}}=\mathrm{K}{2}$
$\frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{K}{1}-\mathrm{K}{2} \mathrm{v}$
$\int_{0}^{v} \frac{d v}{K_{1}-K_{2} v}=\int_{0}^{t} d t$ $-\frac{1}{\mathrm{~K}{2}} \ln \left[\mathrm{K}{1}-\mathrm{K}{2} \mathrm{v}\right]{0}^{\mathrm{v}}=\mathrm{t}$
$\ln \left(\frac{K_{1}-K_{2} v}{K_{1}}\right)=-K_{2} t$
$K_{1}-K_{2} v=K_{1} e^{-K_{2} t}$
$v=\frac{K_{1}}{K_{2}}\left[1-e^{-K_{2} t}\right]$
