If a body moves through a liquid or a gas then the fluid applies a force on the body which is called drag force. Direction of the drag force is always opposite to the motion of the body relative to the fluid. At low speeds of the body, drag frog `( F _(P))` is directly proportional to the speed.
`F_(D) = kv `
What K is a proportionally constant and it depends upon the dimension of the body moving in air at relatively high speeds, the drag force applied by air an the body is proportional to `v^(2)`
Where this proportionally constant K can be given by
`K_(2) rho CA`
Where `rho` is the density of air
C is another constant givig the drag property of air
A is area of cross-section of the body
Consider a case an object of mass m is released from a height h and it falls under gravity. As it's speed increases the drag force starts increasing on the object. Due to this at some instant, the object attains equilibrium. The speed attained by the body at this instant is called "terminal speed" of the body.
Assume that the drag force applied by air on the body follows the relation `F_(D) = kv`,neglect the force by buoyancy applied by air on the body then answer the following questions.
What is the pattern of acceleration change of the body ?

Options

A. It first increases, then decreases to zero
B. It decreases uniformly from starting to zero
C. It decreases non-uniformly from starting to zero
D. It first increases then becomes constant

Answer

To solve the problem, we need to analyze the forces acting on the object as it falls through the air and how these forces affect its acceleration.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Body:**
- The weight of the body (downward force) is given by \( F_g = mg \), where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.
- The drag force acting on the body (upward force) is given by \( F_D = kv \), where \( k \) is a proportionality constant and \( v \) is the velocity of the body.

2. **Write the Equation of Motion:**
- According to Newton's second law, the net force acting on the body can be expressed as:
\[ F_{\text{net}} = F_g - F_D = mg - kv \]
- This net force is equal to the mass of the body multiplied by its acceleration \( a \):
\[ mg - kv = ma \]

3. **Express Acceleration in Terms of Velocity:**
- Rearranging the equation gives us:
\[ ma = mg - kv \]
- Dividing both sides by \( m \):
\[ a = g - \frac{k}{m}v \]
- Let \( c = \frac{k}{m} \), then we can rewrite the equation as:
\[ a = g - cv \]

4. **Analyze the Pattern of Acceleration Change:**
- The equation \( a = g - cv \) indicates that acceleration \( a \) is dependent on the velocity \( v \).
- As the velocity \( v \) increases, the term \( cv \) increases, which means that the acceleration \( a \) decreases.

5. **Determine the Behavior of Acceleration:**
- Initially, when the object is released, \( v = 0 \) and thus \( a = g \).
- As the object falls and gains speed, \( v \) increases, causing \( a \) to decrease.
- Eventually, as \( v \) approaches a certain value, the acceleration will decrease towards zero.

6. **Conclusion on the Pattern of Acceleration Change:**
- The acceleration decreases non-uniformly as the velocity increases. This is because the drag force increases with velocity, leading to a reduction in acceleration.
- Therefore, the acceleration will not decrease at a constant rate but will slow down as it approaches zero.

### Final Answer:
The pattern of acceleration change of the body is that it decreases non-uniformly from the initial value \( g \) to zero as the velocity increases.