Assertion : A particle is projected with velocity u at angle `45^@` with ground. Let v be the velocity of particle at time `(!=0)`, then value of u.v can be zero.
Reason : Value of dot product is zero when angle between two vectors is `90^@`

Options

A. (a)If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. (b)If both Assertion and Reason are true and the Reason is not the correct explanation of the Assertion.
C. (c) If Assertion is true, but the Reason is false.
D. (d) If Assertion is false, but the Reason is true.

Explanation

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

### Step 1: Understand the Assertion
The assertion states that a particle is projected with velocity \( u \) at an angle of \( 45^\circ \) with the ground. We need to determine if the dot product \( u \cdot v \) can be zero at some time \( t \neq 0 \).

### Step 2: Analyze the Motion
In projectile motion, the velocity of the particle can be broken down into two components:
- Horizontal component: \( u_x = u \cos(45^\circ) = \frac{u}{\sqrt{2}} \)
- Vertical component: \( u_y = u \sin(45^\circ) = \frac{u}{\sqrt{2}} \)

At any time \( t \), the velocity \( v \) of the particle can be expressed as:
- \( v_x = u_x = \frac{u}{\sqrt{2}} \) (remains constant)
- \( v_y = u_y - g t = \frac{u}{\sqrt{2}} - g t \)

### Step 3: Determine the Condition for Dot Product to be Zero
The dot product \( u \cdot v \) is given by:
\[ u \cdot v = |u| |v| \cos(\theta) \]
where \( \theta \) is the angle between the vectors \( u \) and \( v \). For the dot product to be zero, \( \cos(\theta) \) must be zero, which occurs when \( \theta = 90^\circ \).

### Step 4: Find When \( \theta = 90^\circ \)
The angle between the initial velocity vector \( u \) and the velocity vector \( v \) at time \( t \) can be determined. As the particle moves, at the peak of its trajectory, the vertical component of the velocity \( v_y \) becomes zero, and the velocity vector \( v \) is purely horizontal. At this point, the angle between \( u \) (which has both horizontal and vertical components) and \( v \) (which is horizontal) is \( 90^\circ \).

### Step 5: Conclusion on Assertion
Since at the peak of the projectile's motion, the angle between \( u \) and \( v \) becomes \( 90^\circ \), the dot product \( u \cdot v \) can indeed be zero at that moment. Thus, the assertion is true.

### Step 6: Analyze the Reason
The reason states that the value of the dot product is zero when the angle between the two vectors is \( 90^\circ \). This is a true statement regarding the properties of the dot product.

### Step 7: Relationship Between Assertion and Reason
While both the assertion and reason are true, the reason does not specifically explain the assertion in the context of projectile motion. The assertion discusses a specific scenario in projectile motion, while the reason discusses a general property of dot products.

### Final Answer
Both the assertion and reason are true, but the reason is not the correct explanation of the assertion. Therefore, the correct answer is **Option B**.

---