64 drops of water, each of radius 2 mm are falling through air at a terminal velocity of 8 cms^

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64 drops of water, each of radius 2 mm are falling through...

64 drops of water, each of radius $2 \mathrm{~mm}$ are falling through air at a terminal velocity of $8 \mathrm{cms}^{-1}$. If they coalesce to form a single drop, then the terminal velocity of combined drop will be: (a) $16 \mathrm{cms}^{-1}$ (b) $128 \mathrm{cms}^{-1}$ (c) $32 \mathrm{cms}^{-1}$ (d) $4 \mathrm{cms}^{-1}$

64 drops of water, each of radius $2 \mathrm{~mm}$ are falling through air at a terminal velocity of $8 \mathrm{cms}^{-1}$. If they coalesce to form a single drop, then the terminal velocity of combined drop will be:
(a) $16 \mathrm{cms}^{-1}$
(b) $128 \mathrm{cms}^{-1}$
(c) $32 \mathrm{cms}^{-1}$
(d) $4 \mathrm{cms}^{-1}$

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Key Concepts

Terminal Velocity under Viscous Forces (Stokes’ Law)

For small spherical droplets moving at low Reynolds numbers in a viscous medium, the terminal velocity is determined by the balance between gravitational force and viscous drag. According to Stokes’ law, the terminal velocity is directly proportional to the square of the drop's radius. Consequently, any increase in the radius leads to a significant increase in the terminal velocity.

Scaling Relationships for Geometric Growth

When additional droplets coalesce, the radius of the resulting drop increases as the cube root of the number of droplets (since volume is proportional to the cube of the radius). This scaling law is fundamental in understanding how the dimensions of a combined drop change relative to the dimensions of the initial droplets.

Volume Conservation in Coalescence

When several droplets merge to form one combined drop, the total volume of the resultant drop equals the sum of the volumes of the individual droplets. As volume is a conserved quantity in this process, it is used to determine the new radius of the drop via the relation V = (4/3)?r³.

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