Which of the following is the graph between the time and the velocity of a body moving in a viscous fluid?

Options

A.
B.
C.
D.

Answer

To determine the graph between time and velocity of a body moving in a viscous fluid, we need to analyze the motion of the body as it falls through the fluid.

### Step-by-Step Solution:

1. **Understanding Forces Acting on the Body**:
- When a body is dropped into a viscous fluid, three primary forces act on it:
- The gravitational force (weight) \( F_g = mg \)
- The buoyant force \( F_b \)
- The viscous drag force \( F_v = 6 \pi \eta r v \)
- Here, \( \eta \) is the viscosity of the fluid, \( r \) is the radius of the body, and \( v \) is the velocity.

2. **Initial Conditions**:
- Initially, when the body is dropped, the gravitational force is greater than the sum of the buoyant force and the viscous force. Thus, the body accelerates downwards.

3. **Velocity Increase**:
- As the body descends, the velocity increases due to the net force acting on it. The net force can be expressed as:
\[ F_{net} = F_g - F_b - F_v \]
- As the velocity increases, the viscous force \( F_v \) also increases.

4. **Reaching Terminal Velocity**:
- Eventually, the body will reach a point where the net force becomes zero. This occurs when the gravitational force is balanced by the sum of the buoyant force and the viscous force:
\[ mg = F_b + F_v \]
- At this point, the body stops accelerating and moves with a constant velocity known as the terminal velocity \( v_t \).

5. **Graphical Representation**:
- The graph of velocity vs. time will show an initial increase in velocity as the body accelerates, followed by a plateau where the velocity remains constant (terminal velocity).
- Therefore, the graph will start from the origin, rise sharply, and then level off as it approaches terminal velocity.

6. **Conclusion**:
- Based on the analysis, the correct graph representing the relationship between time and velocity for a body moving in a viscous fluid is a graph that initially increases and then levels off, indicating the attainment of terminal velocity.

### Correct Option:
- The correct option is **A**.