Statement-I : A projectile is thrown with an initial velocity of `(ahat(i)+bhat(j)) m//s`. If range of projectile is maximum then `a=b`.
Statement-II : In projectile motion, angle of projection is equal to `45^(@)` for maximum range condition.

Options

A. If both assertion `&` Reason are True `&` the Reason is a corrrect explanation of the Asserion.
B. If both Assertion `&` Reason are True but Reason is not correct explanation of the Assertion.
C. If Assertion is Trie but the Reason is False.
D. If both Assertion `&` Reason are false

Answer

To solve the problem, we need to analyze the statements given regarding projectile motion.

### Step 1: Understanding the Components of Initial Velocity
The initial velocity of the projectile is given as \( \vec{u} = a \hat{i} + b \hat{j} \) m/s. Here, \( a \) is the horizontal component (along the x-axis) and \( b \) is the vertical component (along the y-axis).

**Hint:** Recall that the horizontal and vertical components of velocity are crucial in determining the trajectory of the projectile.

### Step 2: Expression for Range of Projectile
The range \( R \) of a projectile launched at an angle \( \theta \) with an initial velocity \( u \) is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
where \( g \) is the acceleration due to gravity.

**Hint:** Remember that the maximum range occurs when \( \sin 2\theta = 1 \).

### Step 3: Condition for Maximum Range
The value of \( \sin 2\theta \) is maximum (equal to 1) when \( 2\theta = 90^\circ \), which implies \( \theta = 45^\circ \).

**Hint:** Think about how the angle of projection affects the range of the projectile.

### Step 4: Components at 45 Degrees
When the angle of projection \( \theta = 45^\circ \), the components of the initial velocity can be expressed as:
\[ u_x = u \cos 45^\circ = \frac{u}{\sqrt{2}}, \quad u_y = u \sin 45^\circ = \frac{u}{\sqrt{2}} \]
This means that the horizontal and vertical components of the initial velocity are equal.

**Hint:** Consider how trigonometric functions behave at \( 45^\circ \).

### Step 5: Relating Components to Statements
From the above, if \( a = u_x \) and \( b = u_y \) at \( 45^\circ \), we find that:
\[ a = b \]
This confirms Statement I, which states that if the range of the projectile is maximum, then \( a = b \).

**Hint:** Reflect on how the equality of components leads to the conclusion about the angle of projection.

### Step 6: Conclusion about Statements
- **Statement I** is true: For maximum range, \( a \) must equal \( b \).
- **Statement II** is also true: The angle of projection for maximum range is indeed \( 45^\circ \).

Thus, both statements are true, and Statement II provides the correct explanation for Statement I.

**Final Conclusion:** Both Statement I and Statement II are true, and Statement II is the correct explanation for Statement I.