Assertion : If velocity - time equation of a particle moving in a straight line is quadratic in time, then displacement - time equation cannot be linear.
Reason : If displacement - time is quadratic in time, then velocity - time is linear.

Options

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

Explanation

To solve the problem, we need to analyze both the assertion and the reason provided in the question.

### Step 1: Understand the Assertion
The assertion states that if the velocity-time equation of a particle moving in a straight line is quadratic in time, then the displacement-time equation cannot be linear.

- **Velocity-Time Equation**: Let's assume the velocity \( v(t) \) is given by a quadratic equation:
\[ v(t) = at^2 + bt + c \]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: Relate Velocity to Displacement
We know that velocity is the derivative of displacement with respect to time:
\[ v(t) = \frac{ds}{dt} \]
If \( v(t) \) is quadratic, we can integrate it to find the displacement \( s(t) \).

### Step 3: Integrate the Velocity Function
Integrating the velocity function:
\[ s(t) = \int v(t) \, dt = \int (at^2 + bt + c) \, dt \]
This gives:
\[ s(t) = \frac{a}{3}t^3 + \frac{b}{2}t^2 + ct + C \]
where \( C \) is the constant of integration.

### Step 4: Analyze the Displacement Equation
From the integration, we see that the displacement \( s(t) \) is a cubic function of time \( t \). A cubic function cannot be linear. Therefore, the assertion is correct: if the velocity-time equation is quadratic, the displacement-time equation cannot be linear.

### Step 5: Understand the Reason
The reason states that if the displacement-time equation is quadratic in time, then the velocity-time equation is linear.

- **Displacement-Time Equation**: Suppose \( s(t) \) is quadratic:
\[ s(t) = At^2 + Bt + D \]
where \( A \), \( B \), and \( D \) are constants.

### Step 6: Differentiate to Find Velocity
To find the velocity, we differentiate the displacement:
\[ v(t) = \frac{ds}{dt} = 2At + B \]
This is a linear function of time \( t \).

### Conclusion
Both the assertion and the reason are correct. However, the reason does not provide a justification for the assertion. Therefore, the correct option is that the assertion is true, and the reason is also true, but the reason is not the correct explanation for the assertion.

### Final Answer
The assertion is true, and the reason is true, but the reason is not the correct explanation for the assertion.

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