Question

A water droplet falls in air and attains terminal velocity \(v_1\). If it splits into \(64\) identical droplets each having terminal velocity \(v_2\). Find \( \dfrac{v_2}{v_1} \).

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{16} \)
• D. \( \frac{1}{32} \)

Answer 1

The Correct Option is C

Concept: Terminal velocity of a small spherical drop in a viscous medium is given by \[ v_T = \frac{2r^2 g}{9\eta}(\rho_s-\rho_f) \] Thus \[ v_T \propto r^2 \]
Step 1:Relate terminal velocity and radius. \[ v \propto r^2 \] \[ \frac{v_2}{v_1} = \left(\frac{r_2}{r_1}\right)^2 \]
Step 2:Use conservation of volume. When the droplet splits into \(64\) identical droplets: \[ \frac{4}{3}\pi r_1^3 = 64 \times \frac{4}{3}\pi r_2^3 \] \[ r_1^3 = 64 r_2^3 \] \[ r_1 = 4r_2 \] \[ \frac{r_2}{r_1} = \frac{1}{4} \]
Step 3:Find velocity ratio. \[ \frac{v_2}{v_1} = \left(\frac{1}{4}\right)^2 \] \[ \frac{v_2}{v_1} = \frac{1}{16} \] \[ \boxed{\frac{1}{16}} \]