64 equal drops of water are falling through air with a steady velocity V0, if the drops coalesce

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64 equal drops of water are falling through air with a...

64 equal drops of water are falling through air with a steady velocity $\mathrm{V}_{0}$, if the drops coalesce, what is their new velocity?
(A) $4 \mathrm{~V}_{0}$
(B) $2^{\{(-1) / 3\}} \mathrm{V}_{0}$
(C) $2 \mathrm{~V}_{0}$
(D) $2^{(1 / 3)} \mathrm{V}_{0}$

Key Concepts

Coalescence and Conservation of Mass

Coalescence refers to the merging of individual droplets into a larger droplet, which changes the overall mass and size while conserving the total mass. When droplets coalesce, the new droplet’s characteristics, such as radius and terminal velocity, must be determined by applying the conservation of mass along with the appropriate scaling laws, which directly impact the dynamics of its motion through the fluid.

Terminal Velocity

Terminal velocity is the steady speed acquired by an object falling through a fluid when the gravitational force is balanced by the drag force. In this state, the acceleration is zero because the net force is zero. This concept is essential when analyzing how changes in mass and size, such as through coalescence, affect the falling speed.

Drag Force and Its Dependence on Size

The drag force experienced by a falling object is influenced by its shape, size, and the properties of the fluid. For spherical objects, the drag can commonly be related to the cross?sectional area, which scales with the square of the radius. This means larger spheres experience greater drag, and understanding this relationship is key to predicting changes in terminal velocity.

Scaling Laws in Fluid Mechanics

Scaling laws relate how physical quantities change with size. In the context of droplets, mass scales with the cube of the radius while the drag (or cross-sectional area) scales with the square. This relationship implies that when droplets coalesce to form a larger droplet, the characteristics of terminal velocity change according to a power law, typically leading to an increase in terminal velocity based on the new radius.

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