To solve the problem of identifying the correct statements regarding a particle moving along a straight line, we will analyze each option based on the principles of motion.
### Step-by-Step Solution:
1. **Understanding the Motion**:
- A particle moving in a straight line can have three key parameters: position, velocity, and acceleration.
- Position indicates where the particle is located, velocity indicates how fast and in which direction it is moving, and acceleration indicates how the velocity is changing.
2. **Analyzing Option A**:
- **Statement**: If the velocity and acceleration have opposite signs, then the object is slowing down.
- **Analysis**: This statement is correct. When the velocity is in one direction and the acceleration is in the opposite direction, the particle's speed decreases, indicating that it is slowing down.
3. **Analyzing Option B**:
- **Statement**: If the position and velocity have opposite signs, then the particle is moving towards the origin.
- **Analysis**: This statement is also correct. If the position is positive (to the right of the origin) and the velocity is negative (to the left), the particle is indeed moving towards the origin. Similarly, if the position is negative and the velocity is positive, the particle is again moving towards the origin.
4. **Analyzing Option C**:
- **Statement**: If the velocity is zero at an instant, then the acceleration should also be zero at that instant.
- **Analysis**: This statement is incorrect. A classic example is a projectile at its highest point where the velocity is zero but the acceleration due to gravity is not zero (it is -9.81 m/s²). Therefore, acceleration can exist even when velocity is zero.
5. **Analyzing Option D**:
- **Statement**: If the velocity is zero for a time interval, then the acceleration must also be zero for that interval.
- **Analysis**: This statement is correct. If the velocity is zero throughout a time interval, the change in velocity is zero, leading to zero acceleration (since acceleration is defined as the change in velocity over time).
6. **Conclusion**:
- The correct options are A, B, and D.
### Final Answer:
The correct statements for a particle going on a straight line are:
- A: If the velocity and acceleration have opposite signs, then the object is slowing down.
- B: If the position and velocity have opposite signs, then the particle is moving towards the origin.
- D: If the velocity is zero for a time interval, then the acceleration must also be zero for that interval.
