Question

Eight equal drops of water are falling through air with a steady speed of 10 cm/s. If the drops coalesce, the new velocity is :

• A. 5 cm/s
• B. 40 cm/s
• C. 16 cm/s
• D. 10 cm/s

Answer 1

The Correct Option is B

To find the new velocity of the coalesced drop, we can use the conservation of volume principle and the equation for terminal velocity of falling objects. When the drops coalesce, they form a single larger drop. Let’s go through the solution step-by-step:

1. Determine the volume of one drop:

   Since all drops are equal in size, the volume of one small drop (V) can be expressed in terms of any variable, but for coalescing purposes, the volume remains constant.

2. Total volume before and after coalescence:

   The total volume of the eight small drops is given by 8V.

   When these drops coalesce, the total volume of the resulting larger drop will still be 8V because volume is conserved.

3. Volume and radius relation:

   The volume of a sphere is given by the formula V = \frac{4}{3} \pi r^3. Thus, the radius of the larger drop is determined by setting its volume to 8V:

   \frac{4}{3} \pi R^3 = 8 \times \frac{4}{3} \pi r^3

   Solving for the new radius R in terms of the original radius r:

   R^3 = 8r^3

   R = 2r

4. Terminal Velocity Proportional to Radius:

   The terminal velocity of a spherical drop in air is proportional to the square of its radius:

   v \propto r^2

   If the original velocity is v = 10 \, \text{cm/s}, the new velocity (V) is given by:

   V = k R^2 and v = k r^2

   Thus, \frac{V}{v} = \left(\frac{R}{r}\right)^2

   Substitute R = 2r:

   \frac{V}{10} = (2)^2 = 4

5. Calculate the new velocity:

   Multiply both sides by 10:

   V = 40 \, \text{cm/s}

Therefore, the correct answer is: 40 cm/s. This explanation utilizes principles of fluid dynamics and continues with mathematical derivation based on given conditions.