On a straight road a thief, seeing a police car approaching at a constant speed v, runs away on a bike with constant accelration 'a'. Which of the following position time graphs best represent the case in which the theif is just nabbed ?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of both the thief and the police car and determine which position-time graph represents the scenario where the thief is just caught by the police.

### Step-by-Step Solution:

1. **Understanding the Motion of the Police Car:**
- The police car is moving at a constant speed \( v \).
- In a position-time graph, constant speed is represented by a straight line with a constant slope.
- Therefore, the graph for the police car will be a straight line.

2. **Understanding the Motion of the Thief:**
- The thief is accelerating from rest with a constant acceleration \( a \).
- The position of an object under constant acceleration can be represented by a parabolic curve in a position-time graph.
- This means that the thief's graph will be a curve that starts at the origin (if we assume he starts from rest) and increases in steepness over time.

3. **Analyzing the Meeting Point:**
- The point at which the thief is caught by the police is where the two graphs intersect.
- This intersection point represents the time and position at which both the police car and the thief are at the same position on the road.

4. **Evaluating the Given Graphs:**
- We need to look for a graph where:
- The police car's graph is a straight line (constant speed).
- The thief's graph is a curve (indicating acceleration).
- The two graphs intersect at one point, indicating the moment the thief is caught.

5. **Conclusion:**
- After evaluating the options, the only graph that meets all these criteria is the one where the police car's straight line intersects with the thief's accelerating curve at a single point.

### Final Answer:
The correct position-time graph is the one where the straight line (representing the police car) intersects the curve (representing the thief) at one point.