The speed ofa ball ofradius 2 cm in a viscous liquid is 20 cm/s. Then the speed of ball of radius I cm in the same liquid is

Options

A. 5 cm/s
B. 8 cm/s
C. 10 cm/s
D. 4 cm/s

Explanation

To solve the problem of finding the speed of a ball of radius 1 cm in a viscous liquid when we know the speed of a ball of radius 2 cm, we can use the relationship between the terminal velocity of a sphere in a viscous fluid and its radius.

### Step-by-Step Solution:

1. **Understand the relationship**: The terminal velocity \( v \) of a sphere in a viscous fluid is proportional to the square of its radius \( r \). This can be expressed mathematically as:
\[ v \propto r^2 \]

2. **Set up the ratio**: From the proportionality, we can write:
\[ \frac{v_1}{v_2} = \frac{r_1^2}{r_2^2} \]
where:
- \( v_1 \) is the speed of the first ball (radius \( r_1 = 2 \) cm),
- \( v_2 \) is the speed of the second ball (radius \( r_2 = 1 \) cm).

3. **Substitute known values**: We know that \( v_1 = 20 \) cm/s and \( r_1 = 2 \) cm, \( r_2 = 1 \) cm. Plugging these values into the equation gives:
\[ \frac{20}{v_2} = \frac{2^2}{1^2} \]

4. **Calculate the squares**: Calculate the squares of the radii:
\[ \frac{20}{v_2} = \frac{4}{1} \]

5. **Cross-multiply to solve for \( v_2 \)**: This leads to:
\[ 20 = 4 v_2 \]

6. **Isolate \( v_2 \)**: Divide both sides by 4:
\[ v_2 = \frac{20}{4} = 5 \text{ cm/s} \]

7. **Conclusion**: The speed of the ball with a radius of 1 cm in the same viscous liquid is:
\[ v_2 = 5 \text{ cm/s} \]

### Final Answer:
The speed of the ball of radius 1 cm in the viscous liquid is **5 cm/s**.