The terminal velocity of a metallic ball of radius 6 text mm$6 \text{ mm}$ in a viscous fluid is 20 text cm/s$20 \text{ cm/s}$. The terminal velocity of another ball of same material and having radius 3 text mm$3 \text{ mm}$ in the same fluid will be ________ cm/s.

### Related Formula v_T = frac2r^2 g9eta (sigma - rho)$$v_T = \frac{2r^2 g}{9\eta} (\sigma - \rho)$$ ### Core Logic For balls of the same material falling through the same fluid, all terms in the terminal velocity equation except the radius r$r$ are constant. Therefore, v_T propto r^2$v_T \propto r^2$. ### Step 1: Setting up the Ratio frac(v_T)_1(v_T)_2 = left(fracr_1r_2right)^2$$\frac{(v_T)_1}{(v_T)_2} = \left(\frac{r_1}{r_2}\right)^2$$ Substituting the given values: r_1 = 6 text mm$r_1 = 6 \text{ mm}$, (v_T)_1 = 20 text cm/s$(v_T)_1 = 20 \text{ cm/s}$ r_2 = 3 text mm$r_2 = 3 \text{ mm}$ ### Step 2: Final Conclusion frac20(v_T)_2 = left(frac63right)^2$$\frac{20}{(v_T)_2} = \left(\frac{6}{3}\right)^2$$ frac20(v_T)_2 = 2^2 = 4$$\frac{20}{(v_T)_2} = 2^2 = 4$$ (v_T)_2 = frac204 = 5 text cm/s$$(v_T)_2 = \frac{20}{4} = 5 \text{ cm/s}$$ ### Pattern Recognition Terminal velocity is directly proportional to the square of the radius. Halving the radius (6 to 3$6 \to 3$) will drop the terminal velocity by a factor of 1/4$1/4$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Mechanical Properties of Fluids