A spherical liquid drop of radius R acquires the terminal velocity v₁ when falls through a gas of viscosity . Now the drop is broken into 64 identical droplets and each droplet acquires terminal velocity v₂ falling through the same gas. The ratio of terminal velocities v₁/v₂ is ________.

Step-by-step solution

The terminal velocity of a spherical drop of radius r falling through a viscous medium is given by Stokes' law: v = 2 9 r^2 ( - ) g From this relation, the terminal velocity is directly proportional to the square of the radius of the drop: v r^2 When the large drop of radius R is broken into 64 identical droplets of radius r , the total volume remains conserved: 4 3 R^3 = 64 4 3 r^3 R^3 = 64 r^3 R = 4r The ratio of the terminal velocities of the original drop and the smaller droplet is: v₁ v₂ = R^2 r^2 Substituting