Which of the following option correctly describes the variation of the speed ` ` and acceleration `'a'` of a point mass falling vertically in a viscous medium that applies a force `F=-kv`, where `'k'` is constant, on the body?
(Graphs are schematic and drawn to scale)

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of a point mass falling vertically in a viscous medium where the drag force is given by \( F = -kv \). Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Forces Acting on the Mass
When a mass \( m \) falls under the influence of gravity and experiences a drag force due to the viscous medium, the forces acting on it are:
- The gravitational force \( F_g = mg \) acting downward.
- The drag force \( F_d = -kv \) acting upward.

### Step 2: Write the Equation of Motion
Using Newton's second law, we can write the equation of motion for the mass:
\[ ma = mg - kv \]
Where:
- \( a \) is the acceleration of the mass.
- \( g \) is the acceleration due to gravity.
- \( v \) is the velocity of the mass.

### Step 3: Rearranging the Equation
Rearranging the equation gives us:
\[ ma = mg - kv \implies a = g - \frac{k}{m}v \]

### Step 4: Analyze the Behavior as Time Progresses
1. **Initial Condition**: At \( t = 0 \), the mass starts from rest, so \( v(0) = 0 \).
2. **As Time Increases**:
- The velocity \( v \) will increase due to the gravitational force.
- The drag force \( kv \) will also increase with \( v \), which will reduce the acceleration \( a \).

### Step 5: Terminal Velocity
Eventually, the mass will reach a terminal velocity \( v_t \) where the acceleration becomes zero:
\[ 0 = mg - kv_t \implies kv_t = mg \implies v_t = \frac{mg}{k} \]
At this point, the forces balance out, and the mass falls with constant velocity.

### Step 6: Behavior of Acceleration
- Initially, the acceleration \( a \) is equal to \( g \) (maximum).
- As the velocity increases, the acceleration decreases due to the increasing drag force.
- At terminal velocity, the acceleration becomes zero.

### Step 7: Graphical Representation
- The graph of velocity \( v \) vs. time \( t \) will show an increasing curve that approaches \( v_t \) asymptotically.
- The graph of acceleration \( a \) vs. time \( t \) will show a decreasing curve starting from \( g \) and approaching zero.

### Conclusion
Based on the analysis, the correct option would be the one that shows:
- Velocity starting from zero and increasing towards terminal velocity.
- Acceleration starting from \( g \) and decreasing towards zero.