The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be _________ cm/s.
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In ratio problems where all other factors are constant, identify the power of the variable ($r$ in this case) and apply it directly to the change factor ($1/2$ radius $\rightarrow 1/4$ velocity).
The problem involves determining the terminal velocity of another metallic ball of the same material but different radius in a viscous fluid. The radius of the first ball is 6 mm and its terminal velocity is 20 cm/s. For the second ball, the radius is 3 mm. Let's find its terminal velocity, \(v_2\). Terminal velocity \(v_t\) in a viscous fluid for a spherical object is given by the formula: \(v_t \propto r^2\) This means \( \frac{v_1}{v_2} = \left( \frac{r_1}{r_2} \right)^2 \) Given \(v_1 = 20 \text{ cm/s}\), \(r_1 = 6 \text{ mm}\), and \(r_2 = 3 \text{ mm}\), substitute into the equation: \(\frac{20}{v_2} = \left(\frac{6}{3}\right)^2\) \(\frac{20}{v_2} = 4\) Solving for \(v_2\): \(v_2 = \frac{20}{4} = 5 \text{ cm/s}\) Thus, the terminal velocity of the second ball is \(5 \text{ cm/s}\). This value fits within the expected range of 5 to 5. Hence, the computed terminal velocity is verified to be correct.
