From amongst the following curves, which one shows the variation of the velocity v with time t for a small sized spherical body falling vertically in a long column of a viscous liquid

Options

A.
B.
C.
D.

Explanation

To determine the variation of the velocity \( v \) with time \( t \) for a small-sized spherical body falling vertically in a long column of a viscous liquid, we can analyze the motion of the body under the influence of gravity and viscous drag.

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Body**:
- When a small spherical body is released in a viscous liquid, two main forces act on it:
- The gravitational force \( F_g = mg \) (downward).
- The viscous drag force \( F_d \) (upward), which opposes the motion.

2. **Newton's Second Law**:
- According to Newton's second law, the net force acting on the body is given by:
\[ F_{net} = F_g - F_d \]
- This net force causes the body to accelerate.

3. **Initial Acceleration**:
- Initially, when the body is released, the gravitational force is much greater than the drag force, leading to a net downward force. Therefore, the body accelerates downward.

4. **Increasing Velocity**:
- As the body falls, its velocity increases, and consequently, the drag force \( F_d \) (which is proportional to the velocity) also increases. This can be expressed as:
\[ F_d = kv \quad \text{(where \( k \) is a constant)} \]

5. **Reaching Terminal Velocity**:
- Eventually, the drag force will equal the gravitational force:
\[ mg = kv_t \]
- At this point, the net force becomes zero, and the body stops accelerating. The velocity at this point is called the terminal velocity \( v_t \).

6. **Graphical Representation**:
- The graph of velocity \( v \) versus time \( t \) will show:
- An initial increase in velocity (which may appear parabolic due to acceleration).
- A plateau where the velocity becomes constant (terminal velocity).

7. **Conclusion**:
- The correct curve that represents this behavior is one that starts from zero, increases initially, and then levels off to a constant value. This corresponds to option B.

### Final Answer:
The curve that shows the variation of the velocity \( v \) with time \( t \) for a small-sized spherical body falling vertically in a long column of a viscous liquid is **option B**.