Mark the correct statements for a particle go ing on a straight line:
(a) If the velocity and acceleration have opposite sign, the object is slowing down.
(b) If the position and velocity have opposite sign, the particle is moving towards the origin.
(c) If the velocity is zero at an instant, the acceleration should also be zero at that instant.
(d) If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval.

Options

A. a, b & c are correct
B. b & d are correct
C. a, b & d are correct
D. all are correct

Answer

To solve the question, we need to analyze each statement one by one and determine whether they are correct or incorrect based on the principles of motion in a straight line.

### Step-by-Step Solution:

1. **Statement (a)**: "If the velocity and acceleration have opposite signs, the object is slowing down."
- **Analysis**: When velocity and acceleration have opposite signs, it means that the object is decelerating. For example, if an object is moving in the positive direction (positive velocity) and the acceleration is negative, the object is slowing down. Therefore, this statement is **correct**.

2. **Statement (b)**: "If the position and velocity have opposite signs, the particle is moving towards the origin."
- **Analysis**: If the position is positive and the velocity is negative, the particle is indeed moving back towards the origin. Conversely, if the position is negative and the velocity is positive, the particle is also moving towards the origin. Thus, this statement is **correct**.

3. **Statement (c)**: "If the velocity is zero at an instant, the acceleration should also be zero at that instant."
- **Analysis**: This statement is not necessarily true. An object can have zero velocity while still experiencing acceleration (e.g., at the peak of a projectile's motion). Therefore, this statement is **incorrect**.

4. **Statement (d)**: "If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval."
- **Analysis**: If the velocity is zero for a time interval, it does not imply that the acceleration is also zero. An object can have zero velocity but still have a non-zero acceleration during that time. For example, an object thrown upwards will have zero velocity at the peak but will have a downward acceleration due to gravity. Hence, this statement is **incorrect**.

### Summary of Correct Statements:
- The correct statements are (a) and (b).

### Final Answer:
- Correct statements: (a) and (b).

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