Assertion : Displacement-time graph is a parabola corresponding to straight line velocity-time graph.
Reason : lf `v= u +at` then `s= ut+ 1/2 at^2`

Options

A. If the both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the given question, we will analyze both the assertion and the reason step by step.

### Step 1: Understand the Assertion
The assertion states that the displacement-time graph is a parabola corresponding to a straight line velocity-time graph.

### Step 2: Analyze the Velocity-Time Graph
A straight line in a velocity-time (VT) graph indicates that the velocity is either constant or changing at a uniform rate (constant acceleration). If the line is horizontal, the velocity is constant. If it has a slope, it indicates constant acceleration.

### Step 3: Determine the Nature of the Displacement-Time Graph
If the velocity is constant (which corresponds to a horizontal line in the VT graph), the displacement-time (DT) graph will be a straight line. This is because displacement is calculated as:
\[ s = ut + \frac{1}{2} a t^2 \]
If acceleration \( a = 0 \) (which is the case for a horizontal line in the VT graph), the equation simplifies to:
\[ s = ut \]
This shows that the displacement increases linearly with time, resulting in a straight line.

If the VT graph has a constant slope (indicating constant acceleration), then the displacement-time graph will be a parabola, as the displacement will depend on the square of time due to the \( \frac{1}{2} a t^2 \) term.

### Step 4: Conclusion on the Assertion
Given that the assertion claims that the displacement-time graph is a parabola corresponding to a straight line velocity-time graph, and we have established that a straight line VT graph (with zero acceleration) results in a straight line DT graph, the assertion is **false**.

### Step 5: Analyze the Reason
The reason states:
\[ v = u + at \]
\[ s = ut + \frac{1}{2} at^2 \]
These are standard equations of motion. They are indeed true and valid for uniformly accelerated motion.

### Step 6: Conclusion on the Reason
The reason is true as it correctly represents the equations of motion.

### Final Conclusion
- **Assertion**: False (displacement-time graph is a straight line for a constant velocity).
- **Reason**: True (the equations of motion are correct).

### Answer
The correct answer is that the assertion is false, but the reason is true.

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