Two stones with radii 1:2 fall from a great height through atmosphere. Their terminal velocities are in the ratio

Options

A. `2:1`
B. `1:4`
C. `4:1`
D. `1:2`

Answer

To solve the problem of finding the ratio of terminal velocities of two stones with radii in the ratio of 1:2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between terminal velocity and radius**:
The terminal velocity \( V \) of a sphere falling through a fluid is given by the formula:
\[ V = \frac{2r^2 (\rho - \rho_0) g}{9 \eta} \]
where \( r \) is the radius of the sphere, \( \rho \) is the density of the sphere, \( \rho_0 \) is the density of the fluid (air), \( g \) is the acceleration due to gravity, and \( \eta \) is the coefficient of viscosity of the fluid.

2. **Express the terminal velocities for both stones**:
Let the radius of the first stone be \( r_1 \) and the radius of the second stone be \( r_2 \). According to the problem, the ratio of their radii is:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]
Therefore, we can express \( r_1 \) as \( r_1 = r \) and \( r_2 = 2r \).

3. **Write the terminal velocities for both stones**:
The terminal velocity for the first stone \( V_1 \) is:
\[ V_1 = \frac{2r_1^2 (\rho - \rho_0) g}{9 \eta} \]
The terminal velocity for the second stone \( V_2 \) is:
\[ V_2 = \frac{2r_2^2 (\rho - \rho_0) g}{9 \eta} \]

4. **Substitute the values of \( r_1 \) and \( r_2 \)**:
Substitute \( r_1 = r \) and \( r_2 = 2r \) into the equations:
\[ V_1 = \frac{2r^2 (\rho - \rho_0) g}{9 \eta} \]
\[ V_2 = \frac{2(2r)^2 (\rho - \rho_0) g}{9 \eta} = \frac{2 \cdot 4r^2 (\rho - \rho_0) g}{9 \eta} = \frac{8r^2 (\rho - \rho_0) g}{9 \eta} \]

5. **Find the ratio of terminal velocities**:
Now, we can find the ratio of the terminal velocities \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\frac{2r^2 (\rho - \rho_0) g}{9 \eta}}{\frac{8r^2 (\rho - \rho_0) g}{9 \eta}} = \frac{2r^2}{8r^2} = \frac{2}{8} = \frac{1}{4} \]

6. **Express the final ratio**:
Therefore, the ratio of terminal velocities \( V_1 : V_2 \) is:
\[ V_1 : V_2 = 1 : 4 \]

### Final Answer:
The terminal velocities of the two stones are in the ratio \( 1 : 4 \).