For a particle moving in a straight line

Options

A. If velocity is negative and acceleration is positive then speed increases.
B. if velocity is positive and acceleration is negative then speed decreases.
C. if velocity is zero at an instant then acceleration must also be zero at that instant.
D. it is possible that speed of a particle is never zero in an interval of time, but average speed is zero.

Answer

To solve the problem regarding the motion of a particle in a straight line and determine which statement is correct, we will analyze each option based on the principles of physics related to velocity and acceleration.

### Step-by-Step Solution:

1. **Understanding the Concepts**:
- **Velocity**: It is the rate of change of displacement and can be positive or negative, indicating the direction of motion.
- **Acceleration**: It is the rate of change of velocity. It can also be positive or negative, indicating whether the velocity is increasing or decreasing.

2. **Analyzing Each Statement**:
- **Statement A**: "If velocity is negative, acceleration is positive, then speed increases."
- If velocity is negative and acceleration is positive, the particle is moving in the negative direction while accelerating in the positive direction. This means the speed (magnitude of velocity) is decreasing. **This statement is incorrect.**

- **Statement B**: "If velocity is positive, acceleration is negative, speed decreases."
- If velocity is positive and acceleration is negative, the particle is moving in the positive direction but slowing down. Hence, the speed is indeed decreasing. **This statement is correct.**

- **Statement C**: "If velocity is zero at an instant, then the acceleration must be zero at that instant."
- This is not necessarily true. For example, when an object is thrown upwards, its velocity becomes zero at the highest point, but the acceleration due to gravity is still acting downwards (non-zero). **This statement is incorrect.**

- **Statement D**: "It is possible when the speed of the particle is never zero in an interval of time, but the average speed is zero."
- Average speed is defined as total distance traveled divided by total time taken. If the speed is never zero, the average speed cannot be zero. **This statement is incorrect.**

3. **Conclusion**:
- The only correct statement among the options provided is **Statement B**: "If velocity is positive, acceleration is negative, speed decreases."