A particle is observed from two frames `S_(1) and S_(2)`. The frame `S_(2)` moves with respect to `S_(1)` with an acceleration a. Let `F_(1) and F_(2)` be the pseudo forces on the particle when seen from `S_(1) and S_(2)` respectively. Which of the following are not possible ?

Options

A. `F_(1) ne 0,F_(2) =0`
B. `F_(1) ne 0, F_(2) ne 0`
C. `F_(1) =0,F_(2) ne 0`
D. `F_(1) = 0,F_(2) =0`

Answer

To solve the problem, we need to analyze the situation involving two reference frames, \( S_1 \) and \( S_2 \), where \( S_2 \) is accelerating with respect to \( S_1 \). We will identify the pseudo-forces acting on a particle observed from both frames and determine which scenario is not possible.

### Step-by-Step Solution:

1. **Understanding the Frames**:
- Let \( S_1 \) be an inertial frame (not accelerating).
- Let \( S_2 \) be a non-inertial frame that is accelerating with respect to \( S_1 \) with an acceleration \( a \).

2. **Identifying the Pseudo-Forces**:
- In an inertial frame like \( S_1 \), the particle experiences no pseudo-force, so \( F_1 = 0 \).
- In the non-inertial frame \( S_2 \), a pseudo-force \( F_2 \) acts on the particle due to the acceleration of the frame. The pseudo-force is equal to \( -ma \), where \( m \) is the mass of the particle.

3. **Case Analysis**:
- **Case 1**: If \( S_1 \) is at rest and \( S_2 \) is accelerating, then:
- \( F_1 = 0 \) (since \( S_1 \) is inertial).
- \( F_2 \neq 0 \) (since \( S_2 \) is non-inertial).
- **Case 2**: If \( S_2 \) is at rest and \( S_1 \) is accelerating, then:
- \( F_1 \neq 0 \) (since \( S_1 \) is non-inertial).
- \( F_2 = 0 \) (since \( S_2 \) is inertial).
- **Case 3**: If both frames are accelerating with respect to each other, then:
- Both \( F_1 \) and \( F_2 \) will be non-zero.
- **Case 4**: If both frames are at rest relative to each other, then both \( F_1 \) and \( F_2 \) can be zero.

4. **Identifying the Impossible Scenario**:
- The question asks which scenario is not possible.
- If \( F_1 = 0 \) (meaning \( S_1 \) is inertial) and \( F_2 = 0 \) (meaning \( S_2 \) is also inertial), this means both frames cannot be accelerating relative to each other. This contradicts the condition that \( S_2 \) is accelerating with respect to \( S_1 \).
- Therefore, the scenario where both \( F_1 \) and \( F_2 \) are zero is not possible.

### Conclusion:
The scenario where both pseudo-forces \( F_1 \) and \( F_2 \) are zero is not possible.