Assertion: In projectile motion, when horizontal range is n times the maximum height, the angle of projection is given by `tan theta=(4)/(n)`
Reason: In the case of horizontal projection the vertical increases with time.

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that in projectile motion, when the horizontal range \( R \) is \( n \) times the maximum height \( H \), the angle of projection \( \theta \) is given by:
\[ \tan \theta = \frac{4}{n} \]

2. **Formulas for Range and Maximum Height**:
The formulas for the horizontal range \( R \) and maximum height \( H \) in projectile motion are:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]
where \( u \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of projection.

3. **Relating Range and Height**:
According to the assertion, we have:
\[ R = nH \]
Substituting the formulas for \( R \) and \( H \):
\[ \frac{u^2 \sin 2\theta}{g} = n \left( \frac{u^2 \sin^2 \theta}{2g} \right) \]

4. **Simplifying the Equation**:
Cancel \( \frac{u^2}{g} \) from both sides (assuming \( u \neq 0 \)):
\[ \sin 2\theta = n \left( \frac{1}{2} \sin^2 \theta \right) \]
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ 2 \sin \theta \cos \theta = \frac{n}{2} \sin^2 \theta \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 4 \cos \theta = n \sin \theta \]
Dividing both sides by \( \sin \theta \) (assuming \( \sin \theta \neq 0 \)):
\[ \frac{4}{n} = \frac{\sin \theta}{\cos \theta} = \tan \theta \]

6. **Conclusion for the Assertion**:
This confirms that the assertion is correct:
\[ \tan \theta = \frac{4}{n} \]

7. **Understanding the Reason**:
The reason states that in the case of horizontal projection, the vertical velocity increases with time. In horizontal projectile motion, the vertical component of velocity is influenced by gravity, which causes it to increase downwards over time.

8. **Conclusion for the Reason**:
This statement is also correct, but it does not directly explain the assertion regarding the relationship between range and height.

### Final Conclusion:
Both the assertion and the reason are correct, but the reason is not a correct explanation of the assertion. Therefore, the answer is that the assertion is true, the reason is true, but the reason is not the correct explanation of the assertion.