1. Use Stokes' law for terminal velocity

For a small solid sphere falling through a viscous fluid, the terminal velocity is

$v_t = \frac{2}{9}\frac{r^2(\rho_s-\rho_f)g}{\eta}$

where:

• $r$ = radius of sphere,
• $\rho_s$ = density of sphere,
• $\rho_f$ = density of fluid,
• $\eta$ = coefficient of viscosity.

So,

$v_t \propto r^2$

since all other quantities remain the same.

2. Terminal velocity of the original sphere

Original radius = $R$

Hence,

$v_1 \propto R^2$

3. Find radius of each smaller sphere

The original sphere is broken into $27$ identical spheres.

Volume of original sphere:

$V = \frac{4}{3}\pi R^3$

If each smaller sphere has radius $r$, then

$27\left(\frac{4}{3}\pi r^3\right)=\frac{4}{3}\pi R^3$

Cancelling common factors,

$27r^3 = R^3$

$r^3 = \frac{R^3}{27}$

$r = \frac{R}{3}$

4. Terminal velocity of each small sphere

Since terminal velocity varies as square of radius,

$v_2 \propto r^2 = \left(\frac{R}{3}\right)^2 = \frac{R^2}{9}$

Thus,

$v_2 = \frac{v_1}{9}$

Therefore,

$\frac{v_1}{v_2} = 9$

5. Check options

• A: $\frac{1}{9}$ ❌
• B: $\frac{1}{27}$ ❌
• C: $27$ ❌
• D: $9$ ✅

So the correct option is D.