Physics Q34 – JEE-MAIN 2026

Home Physics Mechanical Properties of FluidsQuestion #263

A spherical liquid drop of radius

R

acquires the terminal velocity

v_{1}

when falls through a gas of viscosity

η

. Now the drop is broken into 64 identical droplets and each droplet acquires terminal velocity

v_{2}

falling through the same gas. The ratio of terminal velocities

v_{1} / v_{2}

is _______.

Clue 1 of 3: Volume Conservation

When a large drop breaks into smaller droplets, the total volume of the liquid remains constant.

🧠 Full Solution Path

Step 1: Relate the radii of the large drop and small droplets✦ Active

When the large drop breaks into 64 identical smaller droplets, the total volume of the liquid remains constant. Let $R$ be the radius of the large drop and $r$ be the radius of each small droplet.

\frac{4}{3} π R^{3} = 64 \times \frac{4}{3} π r^{3}

R^{3} = 64 r^{3}

R = (64)^{1 / 3} r

R = 4 r

Thus, the radius of a small droplet is $r = \frac{R}{4}$ .

Step 2: Apply Stokes' Law for terminal velocity○ Expand

The terminal velocity $v_{t}$ of a spherical drop of radius $r$ falling through a viscous medium is given by Stokes' Law:

v_{t} = \frac{2 r^{2} (ρ_{p} - ρ_{f}) g}{9 η}

Here, $ρ_{p}$ is the density of the liquid, $ρ_{f}$ is the density of the gas, $g$ is acceleration due to gravity, and $η$ is the viscosity of the gas. Since the liquid, gas, and gravity are the same for both cases, the term $\frac{2 (ρ_{p} - ρ_{f}) g}{9 η}$ is a constant. Therefore, the terminal velocity is directly proportional to the square of the radius of the drop:

v_{t} \propto r^{2}

So, for the large drop, $v_{1} = C R^{2}$ , and for a small droplet, $v_{2} = C r^{2}$ , where $C$ is the constant of proportionality.

Step 3: Calculate the ratio of terminal velocities○ Expand

Now, we can find the ratio $\frac{v_{1}}{v_{2}}$ :

\frac{v_{1}}{v_{2}} = \frac{C R^{2}}{C r^{2}} = \frac{R^{2}}{r^{2}} = {(\frac{R}{r})}^{2}

Substitute the relationship $R = 4 r$ from Step 1:

\frac{v_{1}}{v_{2}} = {(\frac{4 r}{r})}^{2} = (4)^{2} = 16

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Physics Question 34 – JEE-MAIN 2026

Clue 1 of 3: Volume Conservation

🧠 Full Solution Path