Assertion: When `theta = 45^(@) or 135^(@)`, the value of R remains the same, only the sign changes.
Reason: R= `(u^(2)sin2theta)/(g)`

Options

A. Both assertion and reason are true and reason is the correct explanation of assertion
B. Both assetion and reason are true but reason is not the correct explanation of assertion
C. Assertion is true but reason is false
D. Both assertion and reason are false

Answer

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

**Step 1: Understanding the Assertion**
The assertion states that when the angle \( \theta \) is \( 45^\circ \) or \( 135^\circ \), the value of the range \( R \) remains the same, but only the sign changes.

**Step 2: Understanding the Formula for Range**
The formula for the horizontal range \( R \) of a projectile is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where:
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of projection.

**Step 3: Calculate the Range for \( \theta = 45^\circ \)**
Substituting \( \theta = 45^\circ \) into the formula:
\[ R_1 = \frac{u^2 \sin(2 \times 45^\circ)}{g} = \frac{u^2 \sin(90^\circ)}{g} \]
Since \( \sin(90^\circ) = 1 \):
\[ R_1 = \frac{u^2}{g} \]

**Step 4: Calculate the Range for \( \theta = 135^\circ \)**
Now substituting \( \theta = 135^\circ \):
\[ R_2 = \frac{u^2 \sin(2 \times 135^\circ)}{g} = \frac{u^2 \sin(270^\circ)}{g} \]
Since \( \sin(270^\circ) = -1 \):
\[ R_2 = \frac{u^2 \cdot (-1)}{g} = -\frac{u^2}{g} \]

**Step 5: Comparing the Ranges**
Now we compare \( R_1 \) and \( R_2 \):
- \( R_1 = \frac{u^2}{g} \)
- \( R_2 = -\frac{u^2}{g} \)

We see that the magnitudes of \( R_1 \) and \( R_2 \) are the same, but their signs are different.

**Conclusion**
Thus, the assertion is true, and the reason provided is also true as it correctly explains why the assertion holds. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

**Final Answer**
Both assertion and reason are true, and the reason is the correct explanation of the assertion.

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