A body is fired with a velocity of magnitude `sqrt(gR)ltVltsqrt(2gR)` at an angle of `30^@` with the radius vector of the earth. If at the highest point, the speed of the body is `V//4`, the maximum height attained by the body is equal to

Options

A. `(V^(2))/(8g)`
B. `R`
C. `sqrt(2)R`
D. none of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a body fired with a velocity \( v \) at an angle of \( 30^\circ \) with respect to the radius vector of the Earth. We know that the speed at the highest point is \( \frac{v}{4} \). We need to find the maximum height \( h \) attained by the body in terms of \( R \) (the radius of the Earth).

### Step 2: Apply conservation of angular momentum
At the point of launch, the angular momentum \( L \) about the center of the Earth is given by:
\[ L = m v r \sin(30^\circ) \]
Where \( \sin(30^\circ) = \frac{1}{2} \), so we can simplify this to:
\[ L = m v r \cdot \frac{1}{2} = \frac{m v r}{2} \]

### Step 3: Angular momentum at the highest point
At the highest point, the speed of the body is \( \frac{v}{4} \) and the distance from the center of the Earth is \( R + h \). Therefore, the angular momentum at the highest point is:
\[ L = m \left(\frac{v}{4}\right) (R + h) \]

### Step 4: Set the angular momentum equations equal
Since angular momentum is conserved, we can set the two expressions for angular momentum equal to each other:
\[ \frac{m v r}{2} = m \left(\frac{v}{4}\right) (R + h) \]
Here, \( m \) cancels out from both sides:
\[ \frac{v r}{2} = \frac{v}{4} (R + h) \]

### Step 5: Simplify the equation
Now we can simplify the equation:
\[ \frac{r}{2} = \frac{1}{4} (R + h) \]
Multiplying both sides by 4 gives:
\[ 2r = R + h \]

### Step 6: Rearranging to find \( h \)
Now, we can rearrange this equation to solve for \( h \):
\[ h = 2r - R \]

### Step 7: Substitute \( r \) with \( R \)
Since \( r \) is the radius of the Earth, we can substitute \( r \) with \( R \):
\[ h = 2R - R = R \]

### Conclusion
Thus, the maximum height attained by the body is:
\[ h = R \]

### Final Answer
The maximum height attained by the body is equal to \( R \).

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