Which of the following graphs best represents the motion of a raindrop?

Options

A.
B.
C.
D.

Explanation

To determine which graph best represents the motion of a raindrop, we need to analyze the forces acting on the raindrop and how its velocity changes over time.

### Step-by-Step Solution:

1. **Identify the Forces**:
- When a raindrop falls, it experiences two main forces: gravitational force (downward) and drag force (upward).
- The gravitational force acting on the raindrop is given by \( mg \), where \( m \) is the mass of the raindrop and \( g \) is the acceleration due to gravity.
- The drag force can be expressed as \( F_d = \frac{6}{\eta} r v \), where \( \eta \) is the viscosity of the fluid (air), \( r \) is the radius of the raindrop, and \( v \) is its velocity.

2. **Set Up the Equation of Motion**:
- According to Newton's second law, the net force acting on the raindrop can be expressed as:
\[ mg - \frac{6}{\eta} r v = ma \]
- Here, \( a \) is the acceleration of the raindrop, which can be expressed as \( \frac{dv}{dt} \).

3. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ mg - \frac{6}{\eta} r v = m \frac{dv}{dt} \]
- This can be rewritten as:
\[ \frac{dv}{dt} = \frac{mg - \frac{6}{\eta} r v}{m} \]

4. **Separating Variables**:
- We can separate the variables to integrate:
\[ dt = \frac{m}{mg - \frac{6}{\eta} r v} dv \]

5. **Integrating Both Sides**:
- Integrate both sides:
- The left side integrates from \( 0 \) to \( t \).
- The right side integrates from \( 0 \) to \( v \).
- This results in:
\[ t = \int_0^v \frac{m}{mg - \frac{6}{\eta} r v} dv \]

6. **Finding the Velocity Equation**:
- After performing the integration, we find that the velocity \( v \) approaches a terminal velocity \( V_t \) as time increases:
\[ V_t = \frac{mg}{\frac{6}{\eta} r} \]
- The velocity increases exponentially over time until it reaches this terminal velocity.

7. **Graph Representation**:
- The graph of velocity versus time for a raindrop will show an exponential increase, starting from zero and approaching the terminal velocity asymptotically.

8. **Conclusion**:
- Therefore, the graph that best represents the motion of a raindrop is one that shows an exponential increase in velocity over time, which corresponds to option C.