Assertion : Displacement-time equation of two particles moving in a straight line are, `s_1 = 2t - 4t^2` and `s_2 = -2t + 4t^2.` Relative velocity between the two will go on increasing.
Reason : If velocity and acceleration are of same sign then speed will increase.

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 need to analyze the displacement-time equations of the two particles and determine the relative velocity between them. Let's break this down step-by-step.

### Step 1: Write down the displacement equations
The displacement equations for the two particles are given as:
- Particle 1: \( s_1 = 2t - 4t^2 \)
- Particle 2: \( s_2 = -2t + 4t^2 \)

### Step 2: Find the velocities of both particles
To find the velocities, we differentiate the displacement equations with respect to time \( t \).

For Particle 1:
\[ v_1 = \frac{ds_1}{dt} = \frac{d(2t - 4t^2)}{dt} = 2 - 8t \]

For Particle 2:
\[ v_2 = \frac{ds_2}{dt} = \frac{d(-2t + 4t^2)}{dt} = -2 + 8t \]

### Step 3: Calculate the relative velocity
The relative velocity of Particle 1 with respect to Particle 2 is given by:
\[ v_{rel} = v_1 - v_2 \]
Substituting the expressions for \( v_1 \) and \( v_2 \):
\[ v_{rel} = (2 - 8t) - (-2 + 8t) = 2 - 8t + 2 - 8t = 4 - 16t \]

### Step 4: Analyze the relative velocity
Now, we need to determine if the relative velocity \( v_{rel} = 4 - 16t \) is increasing or decreasing over time.

- The term \( -16t \) indicates that as time \( t \) increases, the relative velocity \( v_{rel} \) decreases because the coefficient of \( t \) is negative.

### Step 5: Conclusion about the assertion
Since the relative velocity \( v_{rel} \) is decreasing over time, the assertion that "the relative velocity between the two will go on increasing" is **false**.

### Step 6: Analyze the reason
The reason states: "If velocity and acceleration are of the same sign, then speed will increase."

To check this, we need to consider the acceleration of both particles:

For Particle 1:
\[ a_1 = \frac{dv_1}{dt} = \frac{d(2 - 8t)}{dt} = -8 \]

For Particle 2:
\[ a_2 = \frac{dv_2}{dt} = \frac{d(-2 + 8t)}{dt} = 8 \]

Now, we check the signs of velocity and acceleration:
- For Particle 1, at \( t = 0 \), \( v_1 = 2 \) (positive) and \( a_1 = -8 \) (negative) → different signs.
- For Particle 2, at \( t = 0 \), \( v_2 = -2 \) (negative) and \( a_2 = 8 \) (positive) → different signs.

Since the velocity and acceleration of both particles are not of the same sign, the reason is **true** in general, but it does not explain the assertion.

### Final Answer
- **Assertion**: False
- **Reason**: True
- Therefore, the correct option is: **Assertion is false, but reason is true.**