Question 1: Eight equal drops of oil are falling through air with steady velocity of 0.1 m/s. The drops recombine to form a single drop, what should be the new terminal velocity?

Solution

Given

No of small drops = 8

Velocity of each individual drop = 0.1 m/s

No of large (combined) drops = 1

Find

Terminal velocity of the combined drop

Formulae

(1) Terminal velocity =

(2) Volume of the sphere,

Assumptions: We make a number of assumptions. Let,
 Mass of the small drop = mVolume of the small drop = VRadius of the small drop = r Mass of the combined drop = MVolume of the combined drop = VcRadius of the combined drop = RTerminal velocity of the combined drop = VcT
Comprehension: The eight drops are of the same fluid and having the same density. Therefore, when they combine together, the mass and volume of the larger drop will be, M = 8m and Vc = 8V. However, the radius of the combined drop will NOT be 8r. This is because the volume and radius has not a linear relation.

Solution

For the solution, we need to find the radius R of the combined drop and then put it in the formula 1 to find the terminal velocity of the combined drop. For the calculation of R, we use formula 2.

Put values in formula 2 to find the volume V of the small drop, Similarly, volume of the combined drop,   Now put the value in Vc = 8V as calculated in comprehension.

Take cube root of the eqution

Put this value in Formula (1) for the terminal velocity of the combined larger drop.

But M = 8m, therefore,

On the RHS, the value in bracket is the terminal velocity of the small droplets. So put it,

Where v = 0.1 m/s is the velocity of the small drop given in the text.