# Eight drops of water.each radius 2mm are falling through air at a terminal velocity of 8cm/sec If they coalesce to form a single drop the terminal velocity of the combined drop will be?

$32$ cm/sec

#### Explanation:

For small Reynold number, the viscous force on the small sphere of radius $r$ moving with a velocity $v$ in a fluid of dynamic viscosity $\setminus \mu$ is given by Stoke's law as

${F}_{v} = 6 \setminus \pi \setminus \mu r v$

For terminal or critical velocity ${v}_{c}$ to achieve, the viscous force on the small sphere (density $\setminus {\rho}_{s}$) must be equal to the net weight of body in the fluid of density $\setminus {\rho}_{f}$ hence we have

${F}_{v} = m g - \setminus \textrm{B u o y a n t f \mathmr{and} c e}$

$6 \setminus \pi \setminus \mu r {v}_{c} = \frac{4 \setminus \pi}{3} {r}^{3} \left(\setminus {\rho}_{s} - \setminus {\rho}_{f}\right) g$

${v}_{c} = \setminus \frac{2}{9} \setminus \frac{\left(\setminus {\rho}_{s} - \setminus {\rho}_{f}\right) g {r}^{2}}{\mu}$

${v}_{c} \setminus \propto {r}^{2}$

the above relation shows that the terminal velocity ${v}_{c}$ of the sphere is directly proportional to the square of radius $r$ keeping other parameters constant

Now, 8 identical water drops each of radius ${r}_{1} = 2 \setminus m m$ collapse to form a single big drop of radius ${r}_{2}$

Now, by the conservation of volume, volume of big drop of water (radius ${r}_{2}$) must be equal to the volume of 8 identical water drops $\left({r}_{1} = 2 m m\right)$ then we have

$\setminus \frac{4 \setminus \pi}{3} {r}_{2}^{2} = 8 \setminus \times \setminus \frac{4 \setminus \pi}{3} {r}_{1}^{2}$

${r}_{2} = 2 {r}_{1}$

${r}_{2} = 2 \left(2\right) = 4 \setminus m m$

From, terminal velocity relation (derived above), we know

${v}_{c} \setminus \propto {r}^{2}$

$\setminus \frac{{\left({v}_{c}\right)}_{2}}{\left({v}_{c}\right) 1} = \setminus \frac{{r}_{2}^{2}}{{r}_{1}^{2}}$

$\setminus \frac{{\left({v}_{c}\right)}_{2}}{8} = \setminus \frac{{4}^{2}}{{2}^{2}}$

$\setminus \frac{{\left({v}_{c}\right)}_{2}}{8} = 4$

${\left({v}_{c}\right)}_{2} = 32 \setminus \setminus \textrm{c \frac{m}{\sec}}$