At any instant, the velocity and acceleration of a particle moving along a straight line are v and a. The speed of the particle is increasing if

Options

A. `vgt0,agt0`
B. `vlt0,agt0`
C. `vgt0,alt0`
D. `vgt0,a=0`

Answer

To determine when the speed of a particle is increasing based on its velocity (v) and acceleration (a), we need to analyze the relationship between these two quantities.

### Step-by-Step Solution:

1. **Understanding Speed and Velocity**:
- Speed is the magnitude of velocity. It is always a non-negative quantity.
- If the velocity is positive, the particle is moving in the positive direction; if negative, it's moving in the negative direction.

2. **Role of Acceleration**:
- Acceleration indicates how the velocity of an object is changing over time.
- If acceleration is positive, it means the velocity is increasing. If it is negative, the velocity is decreasing.

3. **Condition for Increasing Speed**:
- For the speed of the particle to increase, the velocity and acceleration must be in the same direction. This means:
- Both velocity (v) and acceleration (a) must be positive, or
- Both must be negative.

4. **Analyzing the Options**:
- **Option 1**: v is positive, a is positive.
- Here, both are positive, so the speed is increasing.
- **Option 2**: v is negative, a is positive.
- Here, velocity is negative and acceleration is positive, meaning the particle is slowing down (speed is decreasing).
- **Option 3**: v is positive, a is negative.
- Here, velocity is positive but acceleration is negative, meaning the particle is slowing down (speed is decreasing).
- **Option 4**: v is positive, a is 0.
- Here, the speed remains constant since acceleration is zero.

5. **Conclusion**:
- The only scenario where the speed of the particle is increasing is in **Option 1**, where both velocity and acceleration are positive.

### Final Answer:
The speed of the particle is increasing if **v is positive and a is positive** (Option 1).

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