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 moving along a straight line is increasing, we need to analyze the relationship between velocity (v) and acceleration (a).

### Step-by-Step Solution:

1. **Understanding Speed and Velocity**:
- Speed is the magnitude of velocity. If the velocity of the particle is positive, the speed is also increasing if the velocity increases.
- If the velocity is negative, the speed is still increasing if the absolute value of the velocity is increasing (i.e., moving further away from zero).

2. **Acceleration Definition**:
- Acceleration (a) is defined as the rate of change of velocity with respect to time, mathematically expressed as:
\[ a = \frac{dv}{dt} \]
- This means that acceleration indicates how the velocity is changing over time.

3. **Condition for Increasing Speed**:
- The speed of the particle is increasing if the velocity and acceleration have the same sign.
- This can be broken down into two cases:
- **Case 1**: If the velocity (v) is positive and acceleration (a) is also positive, the speed is increasing.
- **Case 2**: If the velocity (v) is negative and acceleration (a) is also negative, the speed is increasing (since the particle is moving in the negative direction but speeding up).

4. **Conclusion**:
- Therefore, we can summarize that the speed of the particle is increasing if:
\[ v > 0 \quad \text{and} \quad a > 0 \quad \text{or} \quad v < 0 \quad \text{and} \quad a < 0 \]
- In simpler terms, the speed is increasing if:
\[ v \cdot a > 0 \]