Eight equal drops of water each of radius r=2 mm are falling through air with a terminal veloci

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Eight equal drops of water each of radius r=2 mm are...

Eight equal drops of water each of radius $r=2 \mathrm{~mm}$ are falling through air with a terminal velocity of $16 \mathrm{~cm} / \mathrm{s}$. The eight drops combine to form a big drop. The terminal velocity of big drop is: (a) $16 \mathrm{~cm} / \mathrm{s}$ (b) $32 \mathrm{~cm} / \mathrm{s}$ (c) $64 \mathrm{~cm} / \mathrm{s}$ (d) none of these

Eight equal drops of water each of radius $r=2 \mathrm{~mm}$ are falling through air with a terminal velocity of $16 \mathrm{~cm} / \mathrm{s}$. The eight drops combine to form a big drop. The terminal velocity of big drop is:
(a) $16 \mathrm{~cm} / \mathrm{s}$
(b) $32 \mathrm{~cm} / \mathrm{s}$
(c) $64 \mathrm{~cm} / \mathrm{s}$
(d) none of these

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Key Concepts

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the gravitational force is balanced by the drag or friction force acting against its motion. In fluid dynamics for small spheres, especially in the regime governed by Stokes' law, this velocity depends on factors such as the radius of the sphere, the viscosity of the medium, and the difference in density between the sphere and the fluid.

Stokes' Law

Stokes’ law applies to small spherical objects moving slowly in a viscous fluid, where the drag force is proportional to the radius of the sphere and its velocity. The law is frequently used to determine the terminal velocity of droplets in air by relating the drag force to the sphere’s dimensions and the fluid’s viscosity.

Volume and Radius Scaling

For spherical objects, the volume is proportional to the cube of the radius. This scaling relationship implies that when multiple identical spheres merge to form a new sphere, the new radius is determined by the cube root of the sum of the individual volumes. Additionally, when the terminal velocity is governed by Stokes’ law, it scales with the square of the radius, leading to predictable changes in drop speed upon coalescence.

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