Mark the correct statements for a particle going on a straight line:

Options

A. If the velocity and acceleration have opposite sign, the object is slowing down.
B. If the position andvelocity have opposite sign, the particle is moving towards the origin.
C. If the velocity is ero at an instant, the acceleration should also be zero at that instasnt.
D. If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval.

Answer

To determine the correct statements for a particle moving in a straight line, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Option 1: If the velocity and acceleration have opposite signs, the object is slowing down.**
- **Analysis:** When the velocity and acceleration are in opposite directions, the speed of the object decreases. For example, if an object is moving in the positive direction (positive velocity) and the acceleration is negative, the object will slow down. This is consistent with the equation \( v = u + at \), where \( a \) is negative. Thus, this statement is **correct**.

2. **Option 2: If the position and velocity have opposite signs, the particle is moving towards the origin.**
- **Analysis:** If the position is negative (to the left of the origin) and the velocity is positive (moving to the right), the particle is indeed moving towards the origin. Conversely, if the position is positive and the velocity is negative, the particle is again moving towards the origin. Therefore, this statement is also **correct**.

3. **Option 3: If the velocity is zero at an instant, the acceleration should also be zero at that instant.**
- **Analysis:** This statement is not necessarily true. For example, in the case of a free-falling object, at the highest point, the velocity is zero, but the acceleration due to gravity is not zero (it is \( g \)). Hence, this statement is **incorrect**.

4. **Option 4: If the velocity for a time interval is 0, the acceleration is 0 at any instant within the time interval.**
- **Analysis:** If the velocity is zero for a time interval, it implies that the object is at rest during that interval. For the velocity to remain zero, the acceleration must also be zero during that time. If acceleration were not zero, the velocity would change. Thus, this statement is **correct**.

### Summary of Correct Statements:
- **Correct Statements:** Option 1, Option 2, Option 4
- **Incorrect Statement:** Option 3