A lead shot of 1 mm diameter falls through a long colummn of glycerine. The variation of the velocity v with distance covered (s) is represented by

Options

A.
B.
C.
D.

Explanation

To solve the problem of a lead shot falling through a long column of glycerine, we need to analyze the forces acting on the shot and how they affect its motion. Here’s a step-by-step solution:

### Step 1: Understand the Forces Acting on the Lead Shot
When the lead shot is released, two main forces act on it:
- **Gravitational Force (Weight)**: This force pulls the shot downward and is given by \( F_{\text{gravity}} = mg \), where \( m \) is the mass of the lead shot and \( g \) is the acceleration due to gravity.
- **Viscous Drag Force**: As the shot moves through glycerine, it experiences a drag force due to the viscosity of the fluid. This force opposes the motion and is given by Stokes' law for small particles, \( F_{\text{viscous}} = 6\pi r \eta v \), where \( r \) is the radius of the shot, \( \eta \) is the viscosity of glycerine, and \( v \) is the velocity of the shot.

### Step 2: Analyze the Motion
Initially, when the shot is released, it accelerates due to the gravitational force. As its velocity increases, the viscous drag force also increases. Eventually, the drag force will balance the gravitational force, leading to a terminal velocity where the net force on the shot becomes zero.

### Step 3: Determine the Velocity vs. Distance Relationship
- **Initial Phase**: When the shot is released, it accelerates, and its velocity increases with distance. This phase can be represented by a curve that rises steeply.
- **Terminal Velocity Phase**: Once the drag force equals the gravitational force, the shot reaches a terminal velocity. At this point, the velocity remains constant as the shot continues to fall.

### Step 4: Graphical Representation
The graph of velocity \( v \) versus distance \( s \) will show:
- An initial increasing slope (representing acceleration).
- A plateau where the velocity becomes constant (representing terminal velocity).

### Conclusion
The correct representation of the velocity \( v \) as a function of distance \( s \) is a curve that starts steep and then levels off, indicating that the lead shot accelerates initially and then moves at a constant speed.

### Final Answer
The variation of the velocity \( v \) with distance covered \( s \) is represented by a graph that starts with a steep incline and then levels off, corresponding to the first option given in the problem.

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