A particle is found to be at rest when seen from a frame `S_(1)` and moving with a constant velocity when seen from another frame `S_2`
a) Both the frames are inertial
b) Both the frames are non inertial
c) `S_(1)` is inertial and `S_2` is non inertial
d) `S_(1)` is non inertial and `S_2` is inertial

Options

A. a, b are true
B. c, d are true
C. b, c are true
D. a, d are true

Answer

To solve the problem, we need to analyze the situation described in the question.

### Step-by-Step Solution:

1. **Understanding the Frames**:
- We have two frames of reference: \( S_1 \) and \( S_2 \).
- In frame \( S_1 \), the particle is at rest. This means that the velocity of the particle with respect to \( S_1 \) is zero.
- In frame \( S_2 \), the particle is moving with a constant velocity. This means that the particle has a non-zero velocity with respect to \( S_2 \).

2. **Defining Inertial and Non-Inertial Frames**:
- An **inertial frame** is one in which Newton's first law of motion holds true; that is, an object not subject to any net force will remain at rest or move at a constant velocity.
- A **non-inertial frame** is one that is accelerating or rotating, where fictitious forces may need to be considered to apply Newton's laws.

3. **Analyzing the Particle's Motion**:
- Since the particle is at rest in \( S_1 \), it implies that \( S_1 \) could be an inertial frame (where no net forces act on the particle).
- Since the particle is moving with a constant velocity in \( S_2 \), it suggests that \( S_2 \) could also be an inertial frame if it is moving uniformly relative to \( S_1 \).

4. **Relative Acceleration**:
- The fact that the particle is at rest in \( S_1 \) and moving with constant velocity in \( S_2 \) indicates that both frames could be inertial.
- If both frames are moving with the same constant velocity, then they are both inertial frames.

5. **Conclusion**:
- Since both frames can be considered inertial frames under the given conditions, the correct option is:
- **a) Both the frames are inertial**.