The speed of a projectile at its maximum height is `sqrt3//2` times its initial speed. If the range of the projectile is n times the maximum height attained by it, n is equal to :

Options

A. `(4)/(3)`
B. `2sqrt(3)`
C. `4sqrt(3)`
D. `(3)/(4)`

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given information
We know that the speed of the projectile at its maximum height is \(\frac{\sqrt{3}}{2}\) times its initial speed. Let the initial speed be \(u\).

### Step 2: Determine the angle of projection
At maximum height, the vertical component of the velocity becomes zero, and the horizontal component remains. The horizontal component of the velocity is given by:
\[ u \cos \theta \]
According to the problem, this is equal to \(\frac{\sqrt{3}}{2} u\). Therefore, we can write:
\[ u \cos \theta = \frac{\sqrt{3}}{2} u \]
Dividing both sides by \(u\) (assuming \(u \neq 0\)):
\[ \cos \theta = \frac{\sqrt{3}}{2} \]
From trigonometry, we know that \(\cos 30^\circ = \frac{\sqrt{3}}{2}\). Thus, the angle of projection \(\theta\) is:
\[ \theta = 30^\circ \]

### Step 3: Calculate the range and maximum height
The range \(R\) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
And the maximum height \(H\) is given by:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 4: Substitute \(\theta\) into the formulas
Substituting \(\theta = 30^\circ\):
- For the range:
\[ R = \frac{u^2 \sin 60^\circ}{g} = \frac{u^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{\sqrt{3} u^2}{2g} \]
- For the maximum height:
\[ H = \frac{u^2 \sin^2 30^\circ}{2g} = \frac{u^2 \cdot \left(\frac{1}{2}\right)^2}{2g} = \frac{u^2 \cdot \frac{1}{4}}{2g} = \frac{u^2}{8g} \]

### Step 5: Relate the range to the maximum height
According to the problem, the range \(R\) is \(n\) times the maximum height \(H\):
\[ R = nH \]
Substituting the expressions for \(R\) and \(H\):
\[ \frac{\sqrt{3} u^2}{2g} = n \cdot \frac{u^2}{8g} \]
Cancelling \(u^2\) and \(g\) from both sides (assuming \(u \neq 0\) and \(g \neq 0\)):
\[ \frac{\sqrt{3}}{2} = n \cdot \frac{1}{8} \]
### Step 6: Solve for \(n\)
Multiplying both sides by 8:
\[ n = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3} \]

### Conclusion
Thus, the value of \(n\) is:
\[ \boxed{4\sqrt{3}} \]