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

Options

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

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: Determine the Escape Velocity
The escape velocity \( v_e \) for a planet of radius \( R \) and mass \( M \) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant.

### Step 2: Calculate the Initial Velocity
Since the body is projected with a velocity equal to half the escape velocity, we have:
\[ v = \frac{1}{2} v_e = \frac{1}{2} \sqrt{\frac{2GM}{R}} = \sqrt{\frac{GM}{2R}} \]

### Step 3: Apply Conservation of Energy
We will use the conservation of mechanical energy to find the maximum height \( h \) attained by the body. The total mechanical energy at the surface of the planet (initial) is the sum of kinetic energy (KE) and gravitational potential energy (PE):
- Initial Kinetic Energy (KE):
\[ \text{KE} = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\sqrt{\frac{GM}{2R}}\right)^2 = \frac{1}{2} m \cdot \frac{GM}{2R} = \frac{mGM}{4R} \]
- Initial Potential Energy (PE):
\[ \text{PE} = -\frac{GMm}{R} \]

Thus, the total initial energy \( E_i \) is:
\[ E_i = \frac{mGM}{4R} - \frac{GMm}{R} = \frac{mGM}{4R} - \frac{4mGM}{4R} = -\frac{3mGM}{4R} \]

### Step 4: Energy at Maximum Height
At the maximum height \( h \), the kinetic energy will be zero, and the potential energy will be:
\[ \text{PE} = -\frac{GMm}{R+h} \]
The total energy \( E_f \) at maximum height is:
\[ E_f = 0 - \frac{GMm}{R+h} = -\frac{GMm}{R+h} \]

### Step 5: Set Initial Energy Equal to Final Energy
By conservation of energy, we set \( E_i = E_f \):
\[ -\frac{3mGM}{4R} = -\frac{GMm}{R+h} \]

### Step 6: Solve for \( h \)
Cancelling \( GMm \) from both sides gives:
\[ \frac{3}{4R} = \frac{1}{R+h} \]
Cross-multiplying yields:
\[ 3(R + h) = 4R \]
Thus,
\[ 3h = 4R - 3R = R \implies h = \frac{R}{3} \]

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