A body is projected vertically upwards from the surface of a planet of radius `R` with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is

Options

A. `(R )/(3 ) `
B. `(R )/(2)`
C. `(R )/(4)`
D. `(R )/(5 )`

Explanation

To solve the problem of finding the maximum height attained by a body projected vertically upwards from the surface of a planet with a velocity equal to half the escape velocity, we can follow these steps:

### Step 1: Understand Escape Velocity
The escape velocity \( v_e \) from the surface of a planet is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

### Step 2: Determine the Initial Velocity
The body is projected with a velocity equal to half the escape velocity:
\[ v_0 = \frac{1}{2} v_e = \frac{1}{2} \sqrt{\frac{2GM}{R}} = \sqrt{\frac{GM}{2R}} \]

### Step 3: Apply Conservation of Energy
Using the principle of conservation of mechanical energy, we can equate the total energy at the surface (point A) and at the maximum height (point B).

At point A (surface of the planet):
- Kinetic Energy (KE) at A:
\[ KE_A = \frac{1}{2} mv_0^2 = \frac{1}{2} m \left(\sqrt{\frac{GM}{2R}}\right)^2 = \frac{1}{2} m \cdot \frac{GM}{2R} = \frac{mGM}{4R} \]
- Potential Energy (PE) at A:
\[ PE_A = -\frac{GMm}{R} \]

Total Energy at A:
\[ E_A = KE_A + PE_A = \frac{mGM}{4R} - \frac{GMm}{R} = \frac{mGM}{4R} - \frac{4mGM}{4R} = -\frac{3mGM}{4R} \]

At point B (maximum height):
- Kinetic Energy (KE) at B:
\[ KE_B = 0 \quad (\text{at maximum height, velocity is zero}) \]
- Potential Energy (PE) at B:
\[ PE_B = -\frac{GMm}{R+h} \]

Total Energy at B:
\[ E_B = KE_B + PE_B = 0 - \frac{GMm}{R+h} = -\frac{GMm}{R+h} \]

### Step 4: Set Total Energies Equal
Since energy is conserved, we set \( E_A = E_B \):
\[ -\frac{3mGM}{4R} = -\frac{GMm}{R+h} \]

### Step 5: Solve for \( h \)
Cancelling \( -GMm \) from both sides:
\[ \frac{3}{4R} = \frac{1}{R+h} \]
Cross-multiplying gives:
\[ 3(R + h) = 4R \]
Expanding and rearranging:
\[ 3h = 4R - 3R \]
\[ 3h = R \quad \Rightarrow \quad h = \frac{R}{3} \]

### Conclusion
The maximum height attained by the body is:
\[ h = \frac{R}{3} \]