The diameter of ball A is half of that of B. The ratio of their terminal velocities will be

Options

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

Answer

To find the ratio of the terminal velocities of two balls A and B, given that the diameter of ball A is half that of ball B, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Diameters:**
Let the diameter of ball A be \( d_A \) and the diameter of ball B be \( d_B \). According to the problem, we have:
\[ d_A = \frac{1}{2} d_B \]

2. **Calculate the Radii:**
The radius of ball A, \( r_A \), is half of its diameter:
\[ r_A = \frac{d_A}{2} = \frac{1}{2} \left(\frac{1}{2} d_B\right) = \frac{1}{4} d_B \]
The radius of ball B, \( r_B \), is:
\[ r_B = \frac{d_B}{2} \]

3. **Use the Formula for Terminal Velocity:**
The terminal velocity \( V \) of a sphere falling through a viscous medium is given by:
\[ V \propto r^2 \]
Therefore, we can express the terminal velocities of balls A and B as:
\[ V_A \propto r_A^2 \quad \text{and} \quad V_B \propto r_B^2 \]

4. **Set Up the Ratio of Terminal Velocities:**
The ratio of the terminal velocities can be expressed as:
\[ \frac{V_A}{V_B} = \frac{r_A^2}{r_B^2} \]

5. **Substitute the Radii:**
Substitute the expressions for \( r_A \) and \( r_B \):
\[ \frac{V_A}{V_B} = \frac{\left(\frac{1}{4} d_B\right)^2}{\left(\frac{1}{2} d_B\right)^2} \]

6. **Simplify the Expression:**
Calculate the squares:
\[ \frac{V_A}{V_B} = \frac{\frac{1}{16} d_B^2}{\frac{1}{4} d_B^2} \]
The \( d_B^2 \) terms cancel out:
\[ \frac{V_A}{V_B} = \frac{1/16}{1/4} = \frac{1}{16} \times \frac{4}{1} = \frac{4}{16} = \frac{1}{4} \]

7. **Final Result:**
Thus, the ratio of the terminal velocities of balls A and B is:
\[ \frac{V_A}{V_B} = \frac{1}{4} \]

### Conclusion:
The ratio of the terminal velocities of ball A to ball B is \( \frac{1}{4} \).

---