A body is projected vertically upwards from the surface of earth with a velocity equal to half the escape velocity. If R be the radius of earth, maximum height attained by the body from the surface of earth is `( R)/(n)`. Find the value of n.

Answer

To solve the problem, we will use the principle of conservation of mechanical energy. The body is projected upwards with a velocity equal to half the escape velocity, and we need to find the maximum height attained by the body from the surface of the Earth.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( v_e \) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface of the Earth and \( R \) is the radius of the Earth.

2. **Initial Velocity**:
Since the body is projected with a velocity equal to half the escape velocity, we have:
\[ v = \frac{1}{2} v_e = \frac{1}{2} \sqrt{2gR} \]

3. **Initial Kinetic Energy**:
The initial kinetic energy \( K_i \) of the body is given by:
\[ K_i = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\frac{1}{2} \sqrt{2gR}\right)^2 = \frac{1}{2} m \cdot \frac{1}{4} (2gR) = \frac{mgR}{4} \]

4. **Initial Potential Energy**:
The initial potential energy \( U_i \) at the surface of the Earth is:
\[ U_i = -\frac{GMm}{R} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

5. **Final Kinetic Energy and Potential Energy**:
At the maximum height \( h \), the final kinetic energy \( K_f \) is 0 (the body momentarily comes to rest), and the potential energy \( U_f \) at height \( h \) is:
\[ U_f = -\frac{GMm}{R + h} \]

6. **Applying Conservation of Energy**:
According to the conservation of mechanical energy:
\[ K_i + U_i = K_f + U_f \]
Substituting the values we have:
\[ \frac{mgR}{4} - \frac{GMm}{R} = 0 - \frac{GMm}{R + h} \]

7. **Simplifying the Equation**:
Rearranging gives:
\[ \frac{mgR}{4} - \frac{GMm}{R} = -\frac{GMm}{R + h} \]
Canceling \( m \) from both sides:
\[ \frac{gR}{4} - \frac{GM}{R} = -\frac{GM}{R + h} \]

8. **Substituting \( g \)**:
We know \( g = \frac{GM}{R^2} \), substituting this into the equation:
\[ \frac{GM}{R^2} \cdot \frac{R}{4} - \frac{GM}{R} = -\frac{GM}{R + h} \]
Simplifying gives:
\[ \frac{GM}{4R} - \frac{GM}{R} = -\frac{GM}{R + h} \]
\[ \frac{GM}{4R} - \frac{4GM}{4R} = -\frac{GM}{R + h} \]
\[ -\frac{3GM}{4R} = -\frac{GM}{R + h} \]

9. **Cross Multiplying**:
Cross multiplying gives:
\[ 3G(R + h) = 4GR \]
Simplifying:
\[ 3R + 3h = 4R \]
\[ 3h = R \]
\[ h = \frac{R}{3} \]

10. **Finding \( n \)**:
From the problem, we have \( h = \frac{R}{n} \). Since we found \( h = \frac{R}{3} \), we can equate:
\[ \frac{R}{n} = \frac{R}{3} \]
Thus, \( n = 3 \).

### Final Answer:
The value of \( n \) is \( 3 \).