In which of the following velocity (v) - time (1) graphs, the instantaneous acceleration decreases with time?

Options

A.
B.
C.
D.

Answer

To determine in which of the given velocity-time graphs the instantaneous acceleration decreases with time, we need to analyze the slopes of the graphs. The slope of a velocity-time graph represents the acceleration. If the slope is decreasing, it indicates that the acceleration is also decreasing.

### Step-by-Step Solution:

1. **Understanding Instantaneous Acceleration**:
- Instantaneous acceleration is defined as the rate of change of velocity with respect to time, mathematically expressed as \( a = \frac{dv}{dt} \).
- In a velocity-time graph, the slope of the graph at any point gives the instantaneous acceleration.

2. **Analyzing Each Option**:
- **Option 1**: The graph shows a parabolic increase in velocity.
- As time increases, the slope of the graph (and thus the acceleration) is increasing. Therefore, the instantaneous acceleration is increasing, not decreasing.

- **Option 2**: The graph shows a linear decrease in velocity.
- The slope is constant and negative, indicating constant negative acceleration. Since the acceleration is constant, it is neither increasing nor decreasing.

- **Option 3**: The graph shows a parabolic decrease in velocity.
- Here, as time increases, the slope of the graph decreases. This indicates that the instantaneous acceleration is decreasing as time progresses. This is a candidate for our answer.

- **Option 4**: The graph shows a straight line increasing in velocity.
- The slope is constant and positive, indicating constant positive acceleration. Therefore, the instantaneous acceleration is constant and not decreasing.

3. **Conclusion**:
- Based on the analysis, the only graph where the instantaneous acceleration decreases with time is **Option 3**.

### Final Answer:
**The instantaneous acceleration decreases with time in Option 3.**