A car starts from erest accelerates uniform by for 4 second and then moves with uniform velocity which of the x-t graph represent th motion of the car:-

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which x-t graph represents the motion of a car that starts from rest, accelerates uniformly for 4 seconds, and then moves with uniform velocity, we can break down the motion into two distinct phases:

### Step 1: Understand the Motion Phases
1. **Phase 1 (0 to 4 seconds)**: The car starts from rest and accelerates uniformly.
- Initial velocity (u) = 0 m/s
- Acceleration (a) = constant
- Time (t) = 4 seconds

2. **Phase 2 (after 4 seconds)**: The car moves with a uniform velocity (constant speed).
- After 4 seconds, the car will continue to move at the velocity it reached at the end of the acceleration phase.

### Step 2: Analyze Phase 1
- Since the car starts from rest and accelerates uniformly, the displacement (x) during this phase can be described by the equation:
\[ x = ut + \frac{1}{2} a t^2 \]
Given that \(u = 0\), the equation simplifies to:
\[ x = \frac{1}{2} a t^2 \]
- The graph of displacement (x) versus time (t) will be a parabola opening upwards, starting from the origin (0,0) since the car starts from rest.

### Step 3: Analyze Phase 2
- After 4 seconds, the car moves with a constant velocity (v). The displacement during this phase can be represented as:
\[ x = vt + C \]
where \(C\) is the displacement at \(t = 4\) seconds.
- The graph of displacement (x) versus time (t) during this phase will be a straight line with a constant slope (the slope represents the constant velocity).

### Step 4: Combine Both Phases
- The overall x-t graph will consist of:
- A parabolic curve from \(t = 0\) to \(t = 4\) seconds (Phase 1).
- A straight line with a constant slope starting from the end of the parabola at \(t = 4\) seconds (Phase 2).

### Step 5: Identify the Correct Graph
- The correct x-t graph will show:
- A curve that starts at the origin and curves upwards for the first 4 seconds.
- After 4 seconds, the graph will transition into a straight line that continues to rise at a constant slope.

Based on this analysis, the correct answer is option **D**, which represents the described motion.