Question

If the ratio of radii of two balls be 2 ∶ 1 then the ratio of their terminal velocities will be - 
1. 1 ∶ 4
2. 4 ∶ 1
3. 1 ∶ 2
4. 2 ∶ 1

Answer 1

Correct Answer - Option 2 : 4 ∶ 1

Concept:

Terminal velocity:

• If a spherical body of radius r is dropped in a viscous fluid, it is first accelerated, and then its acceleration becomes zero and it attains a constant velocity called terminal velocity.
•  The terminal velocity is given as,

\(⇒ v=\frac{2r^2(ρ-σ)}{9η}\)

Where v = terminal velocity, r = radius, ρ = density of the body, σ = density of liquid, g = gravitational acceleration, and η = viscosity

Calculation:

• We know that the terminal velocity is given as,

\(⇒ v=\frac{2r^2(ρ-σ)}{9η}\)

If all quantities other than r is constant, then,

⇒ v ∝ r2 

or

r² ∝ v

⇒ r ² = k v -- (1)

Here, k is constant. 

So, if for radius r₁ the terminal velocity is v₁, and for radius r₂ the terminal velocity is v₂ then we can say

\(\frac{(r_1)^2}{(r_2)^2} = \frac{v_1}{v_2}\) -- (2)

Given, the ratio of radii is 2: 1

\(\frac{(r_1)}{(r_2)} = \frac{2}{1}\) -- (3)

Putting (3) in (2)

\(\frac{(2)^2}{(1)^2} = \frac{v_1}{v_2}\)

\(\implies \frac{4}{1} = \frac{v_1}{v_2}\)

So, the ratio of velocities is 4 : 1.

The correct option is 4 : 1.