A small sphere of mass m is dropped from a aheight After it has fallen 100 m it has attained its terminal velocity and continues to fall at that speed. The work done by air friction against the sphere during the first 100 m of fall is-

Options

A. greater than the work done by air friction in the second 100 m
B. less than the work done by air friction in the second 100 m
C. Equal to 100 mg
D. greater than 100 mg

Answer

To solve the problem of the work done by air friction against a sphere during its first 100 meters of fall, we can follow these steps:

### Step-by-step Solution:

1. **Understanding Terminal Velocity**:
- When an object falls through a fluid (like air), it accelerates until the force of air resistance (drag) equals the gravitational force acting on it. At this point, it reaches terminal velocity and continues to fall at a constant speed.

2. **Identify Forces Acting on the Sphere**:
- The forces acting on the sphere are:
- Gravitational force (weight) \( F_g = mg \) (where \( g \) is the acceleration due to gravity).
- Air resistance (drag force) \( F_d \), which increases with the velocity of the sphere.

3. **Work Done by Air Friction**:
- Work done by air friction can be calculated using the formula:
\[ W = F_d \cdot d \]
- Where \( d \) is the distance fallen (100 m in this case).

4. **Behavior of Air Resistance**:
- During the initial phase of the fall (the first 100 m), the sphere is accelerating and has not yet reached terminal velocity. Therefore, the air resistance is less than the weight of the sphere.
- The work done by air friction during this phase will be less than the work done if the sphere had already reached terminal velocity.

5. **Conclusion**:
- Since the work done by air friction during the first 100 m is less than the work done if the sphere had been falling at terminal velocity, we can conclude that the work done by air friction is a certain value that is less than the maximum possible work done at terminal velocity.

6. **Selecting the Correct Option**:
- Based on the analysis, we can conclude that the work done by air friction during the first 100 m of fall is less than the work done by air friction when the sphere is at terminal velocity. Therefore, the correct option is B.

### Summary:
The work done by air friction against the sphere during the first 100 m of fall is less than the work done at terminal velocity. Thus, the answer is option B.