Question 52: Properties of Matter

Question

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?

Accepted Answer

$$\begin{aligned}
& \mathrm{mg}-\mathrm{F}_{\mathrm{B}}-\mathrm{F}_{\mathrm{v}}=\mathrm{ma} \
& \left(ho \frac{4}{3} \pi \mathrm{r}^3ight) g-\left(ho_{\mathrm{L}} \frac{4}{3} \pi \mathrm{r}^3ight) g-6 \pi \eta r v=m \frac{d v}{d t}
\end{aligned}$$
Let $$\frac{4}{3 m} \pi R^3 g\left(ho-ho_Light)=K_1$$ and $$\frac{6 \pi \eta r}{m}=K_2$$
$$\frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{K}_1-\mathrm{K}_2 \mathrm{~V}$$
$$\begin{aligned}
& \int_\limits0^v \frac{d v}{K_1-K_2 v}=\int_\limits0^t d t \
& -\frac{1}{K_2} \ln \left[K_1-K_2 vight]_0^v=t \
& \ell n\left(\frac{K_1-K_2 v}{K_1}ight)=-K_2 t \
& K_1-K_2 v=K_1 e^{-K_2 t} \
& v=\frac{K_1}{K_2}\left[1-e^{-K_2 t}ight]
\end{aligned}$$

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Properties of Matter

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?

Choose the correct answer: