Two objects A and B are moving with velocities `v_(A) and v_(B)` respectively along positive x-axis. If `v_(A) lt v_(B)` then, which of the following position-time graphs is correctly showing the velocity of A and B ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the position-time graphs of two objects A and B, given that their velocities are \( v_A \) and \( v_B \) respectively, and that \( v_A < v_B \). This means that object A is moving slower than object B.

### Step-by-Step Solution:

1. **Understanding Velocity from Position-Time Graphs**:
- The velocity of an object can be determined from the slope of its position-time (x-t) graph. The slope of the graph is defined as the change in position (Δx) over the change in time (Δt), which is mathematically expressed as:
\[ \text{Velocity} = \frac{\Delta x}{\Delta t} \]
- A steeper slope indicates a higher velocity, while a gentler slope indicates a lower velocity.

2. **Analyzing the Given Condition**:
- We know from the problem statement that \( v_A < v_B \). This means that the slope of the position-time graph for object A must be less than the slope for object B.

3. **Examining Each Graph Option**:
- **Graph 1**: Both curves have the same slope. This implies \( v_A = v_B \), which contradicts our condition \( v_A < v_B \). Thus, this graph is incorrect.
- **Graph 2**: Similar to Graph 1, both curves have the same slope, indicating \( v_A = v_B \). This is also incorrect.
- **Graph 3**: Here, the slope of object B's graph is steeper than that of object A's graph. This indicates \( v_B > v_A \), which satisfies our condition \( v_A < v_B \). Therefore, this graph is a potential correct answer.
- **Graph 4**: The graph for object A has a negative slope, indicating that it is moving in the opposite direction, which contradicts the problem's condition that both objects are moving along the positive x-axis. Thus, this graph is also incorrect.

4. **Conclusion**:
- The only graph that correctly represents the condition \( v_A < v_B \) is Graph 3, where the slope of the graph for object B is greater than that for object A.

### Final Answer:
The correct position-time graph that shows \( v_A < v_B \) is **Graph 3**.

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