The question given below consist of an assertion and the reason . Use the following key to choose appropriate answer:
Assertion: Range of projectile motion is same when particle is projected at an angle `theta` with the horizontal or at an angle `(90^(@) -theta)` with the horizontal .
Reason : Range of projectile motion depended only on the angle of projection.

Options

A. If both assertion and reason are correct and reason is a correct explanation of the assertion
B. If both assertion and reason are correct but reason is not the correct explanation of assertion .
C. If assertion is correct but reason is incorrect .
D. If both assertion and reason are incorrect .

Answer

To solve the question regarding the assertion and reason in projectile motion, we will analyze both statements step by step.

### Step 1: Understanding the Assertion
The assertion states that the range of projectile motion is the same when a particle is projected at an angle θ with the horizontal and at an angle (90° - θ) with the horizontal.

**Hint:** Recall the formula for the range of projectile motion and how it relates to the angle of projection.

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

**Hint:** Remember that \( \sin(2\theta) \) can be expressed in terms of \( \sin \) and \( \cos \).

### Step 3: Evaluating the Range at (90° - θ)
Now, if we project the same particle at an angle (90° - θ), we can find the range using the same formula:
\[ R = \frac{u^2 \sin(2(90° - \theta))}{g} \]
Using the identity \( \sin(90° - x) = \cos(x) \), we have:
\[ \sin(2(90° - \theta)) = \sin(180° - 2\theta) = \sin(2\theta) \]
Thus, the range becomes:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]

**Hint:** Consider how the sine function behaves with angles that sum to 180°.

### Step 4: Conclusion on the Assertion
Since both angles \( \theta \) and (90° - θ) yield the same range, the assertion is correct.

### Step 5: Understanding the Reason
The reason states that the range of projectile motion depends only on the angle of projection. This statement is partially correct because the range also depends on the initial speed \( u \).

**Hint:** Think about the factors that influence the range of a projectile.

### Step 6: Conclusion on the Reason
The range does depend on both the angle of projection and the initial speed. Therefore, the reason is incorrect.

### Final Conclusion
- **Assertion:** True
- **Reason:** False

Thus, the correct answer is that only the assertion is correct.

### Summary of Steps
1. Analyze the assertion about the range of projectile motion.
2. Use the range formula to evaluate the range at angles θ and (90° - θ).
3. Conclude that the assertion is correct.
4. Analyze the reason and conclude it is incorrect.
5. Finalize the answer based on the evaluation of both statements.