Eight identical spherical raindrops are falling through the air with a steady terminal velocity v₀ . If they coalesce to form a single large spherical drop, what will be the new terminal velocity of the large drop?

Step-by-step solution

Let the radius of each small raindrop be r . When 8 identical drops coalesce to form a single large drop of radius R , the total volume remains conserved: 8 4 3 r^3 = 4 3 R^3 R^3 = 8r^3 R = 2r The terminal velocity of a spherical drop of radius r falling through a fluid is given by v_t r^2 . For the initial small drop: v₀ r^2 For the newly formed large drop: v' R^2 Taking the ratio of the two terminal velocities: v' v₀ = ( R r )^2 = ( 2r r )^2 = 4 Thus, the new terminal velocity is v' = 4v₀ . Answer: 4v₀