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Two rain drops of radius r and 2r are falling through air. Find the ratio of their terminal velocities.
1. 1 : 4
2. 1 : 2
3. 1 : 1
4. 2 : 1

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Best answer
Correct Answer - Option 1 : 1 : 4

CONCEPT:

  • Terminal velocity: It is the maximum velocity that an object attained when it falls through a fluid.
    • It occurs when the sum of the viscous force (Fv) and the buoyancy (FB) is equal to the downward force of gravity (Fg) acting on the object.

Fg = FB + Fv

mg = FB + Fv

\(\frac{4}{3}\pi r^3 ρ g = \frac{4}{3}\pi r^3 σ g - 6 \pi η rv\)

\(v=\frac{2}{9} r^2g \left ( \frac{σ - ρ}{η} \right )\)

where v is the terminal velocity, r is the radius of the object, g is the gravitational acceleration, ρ and σ are the density of the medium and object, and η is the viscosity of the medium.

CALCULATION:

Given that radius are r and 2r. So \( {r_1 \over r_2}= {1 \over 2}\)

Terminal velocity v is given by:

\(v=\frac{2}{9} r^2g \left ( \frac{σ - ρ}{η} \right )\)

v α r2 (since all other parameters are the same for both the drops)

\({v_1 \over v_2} = \frac{r_1^2}{r_2^2}= \frac{1^2}{2^2}= \frac{1}{4}\)

So the correct answer is option 1.

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