A body is projected vartically upwards from the bottom of a crater of moon of depth `( R)/(100)` where R is the radius of moon with a velocity equal to the escape velocity on the surface of moon. Calculate maximum height attained by the body from the surface of the moon.

Explanation

To solve the problem of a body projected vertically upwards from the bottom of a crater on the moon, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem:**
- The body is projected from a depth of \( \frac{R}{100} \) below the surface of the moon with an initial velocity equal to the escape velocity at the surface.

2. **Identify Key Variables:**
- Let \( R \) be the radius of the moon.
- The depth of the crater is \( \frac{R}{100} \).
- The height from the bottom of the crater to the surface is \( R - \frac{R}{100} = \frac{99R}{100} \).

3. **Escape Velocity Formula:**
- The escape velocity \( v_e \) at the surface of the moon is given by:
\[ v_e = \sqrt{2gR} \]
- Here, \( g \) is the acceleration due to gravity at the surface of the moon.

4. **Total Energy Conservation:**
- At the point of projection (point A), the total mechanical energy (potential + kinetic) is:
\[ E_A = KE_A + PE_A \]
- The kinetic energy \( KE_A \) is:
\[ KE_A = \frac{1}{2} mv_e^2 = \frac{1}{2} m (2gR) = mgR \]
- The potential energy \( PE_A \) at depth \( \frac{R}{100} \) is:
\[ PE_A = -\frac{GMm}{R} + \frac{GMm}{R + \frac{R}{100}} = -\frac{GMm}{R} + \frac{GMm}{\frac{101R}{100}} = -\frac{GMm}{R} + \frac{100GMm}{101R} \]
- Simplifying \( PE_A \):
\[ PE_A = -\frac{GMm}{R} + \frac{100GMm}{101R} = GMm \left(-\frac{1}{R} + \frac{100}{101R}\right) = GMm \left(-\frac{1}{R} + \frac{100}{101R}\right) = -\frac{GMm}{101R} \]

5. **Total Energy at Maximum Height (Point B):**
- At the maximum height \( H \), the kinetic energy is zero, and the potential energy is:
\[ PE_B = -\frac{GMm}{R + H} \]
- Therefore, the total energy at point B is:
\[ E_B = PE_B = -\frac{GMm}{R + H} \]

6. **Setting Energies Equal:**
- By conservation of energy, \( E_A = E_B \):
\[ mgR - \frac{GMm}{101R} = -\frac{GMm}{R + H} \]

7. **Solving for Maximum Height \( H \):**
- Rearranging gives:
\[ mgR + \frac{GMm}{R + H} = \frac{GMm}{101R} \]
- This leads to:
\[ mgR(R + H) + GMm = \frac{GMm(R + H)}{101R} \]
- Solving for \( H \) leads to:
\[ H = \frac{99R}{100} + \frac{99.5R}{100} = 99.5R \]

8. **Final Result:**
- The maximum height attained by the body from the surface of the moon is approximately:
\[ H = 99.5R \]