To determine which statements are correct, we will analyze each statement one by one based on the principles of motion along a straight line.
### Step-by-Step Solution:
1. **Statement 1: Average speed of a particle in a given time is never less than the magnitude of the average velocity.**
- **Explanation:** Average speed is defined as the total distance traveled divided by the total time taken. Average velocity is defined as the total displacement divided by the total time taken. Since distance (total path length) is always greater than or equal to displacement (straight line from start to end), the average speed will always be greater than or equal to the average velocity.
- **Conclusion:** This statement is **correct**.
2. **Statement 2: It is possible to have a situation in which \( \frac{dv}{dt} \neq 0 \) but \( \frac{d}{dt} |v| = 0 \).**
- **Explanation:** This situation can occur in uniform circular motion. In this case, the speed (magnitude of velocity) remains constant, hence \( \frac{d}{dt} |v| = 0 \). However, the direction of the velocity vector is changing, which means that \( \frac{dv}{dt} \neq 0 \) because the velocity vector is changing in direction even though its magnitude remains constant.
- **Conclusion:** This statement is **correct**.
3. **Statement 3: The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval.**
- **Explanation:** Consider a particle moving in a circular path. If it returns to its starting point, the total displacement is zero, hence the average velocity is zero. However, if the particle is moving continuously without stopping, its instantaneous velocity is never zero at any point during the motion.
- **Conclusion:** This statement is **correct**.
4. **Statement 4: The average velocity of a particle moving on a straight line is zero in a time interval. Instantaneous velocity is never zero in the interval.**
- **Explanation:** For the average velocity to be zero, the displacement must be zero. If a particle moves from point A to point B and back to point A, the average velocity is zero. However, during this motion, the particle must have an instantaneous velocity at various points, including when it changes direction, which means it will be zero at some point (when it turns around).
- **Conclusion:** This statement is **incorrect**.
### Final Conclusion:
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Correct
- Statement 4: Incorrect
Thus, the correct options are **1, 2, and 3**.
