A small tiny lead shot is gently dropped on the surface of a viscous liquid

Options

A. the lead shot will fall will an acceleration equal to g at that place
B. the velocity of the lead shot will decrease with time
C. the velocity of the lead shot will increase continuously
D. the velocity of the lead shot will reach a steady value after sometime

Answer

To solve the problem of a small lead shot being dropped into a viscous liquid, we need to analyze the forces acting on the lead shot and understand the concept of terminal velocity.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Lead Shot:**
- When the lead shot is dropped, two main forces act on it:
- **Weight (mg)**: This force acts downward due to gravity, where \( m \) is the mass of the lead shot and \( g \) is the acceleration due to gravity.
- **Drag Force (F_drag)**: This is the upward force exerted by the viscous liquid opposing the motion of the lead shot.

2. **Set Up the Equation of Motion:**
- According to Newton's second law, the net force acting on the lead shot can be expressed as:
\[ F_{net} = mg - F_{drag} \]
- The acceleration \( a \) of the lead shot can thus be expressed as:
\[ ma = mg - F_{drag} \]
- Rearranging gives:
\[ a = g - \frac{F_{drag}}{m} \]

3. **Understanding Terminal Velocity:**
- As the lead shot falls, it accelerates until the drag force increases to a point where it equals the weight of the lead shot:
\[ F_{drag} = mg \]
- At this point, the net force becomes zero, and the acceleration becomes zero. The lead shot then moves with a constant velocity known as terminal velocity \( v_t \).

4. **Analyze the Options Given:**
- **Option 1**: The lead shot will fall with an acceleration equal to \( g \).
- **Incorrect**: The presence of drag force means that the acceleration will be less than \( g \).
- **Option 2**: The velocity of the lead shot will decrease with time.
- **Incorrect**: The velocity will initially increase until terminal velocity is reached.
- **Option 3**: The velocity of the lead shot will increase continuously.
- **Incorrect**: The velocity will increase until it reaches terminal velocity, after which it will remain constant.
- **Option 4**: The velocity of the shot will reach a steady value after some time.
- **Correct**: This is true as the lead shot will eventually reach terminal velocity.

### Conclusion:
The correct answer is that the velocity of the lead shot will reach a steady value after some time due to the balance between the weight and the drag force.