A projectile is fired from the surface of earth of radius `R` with a velocity `k upsilon_(e)` (where `upsilon_(e)` is the escape velocity from surface of earth and `k lt 1)`. Neglecting air resistance, the maximum height of rise from centre of earth is

Options

A. `(a)(R )/(k^(2) - 1)`
B. `(b)k^(2) R`
C. `(c)(R )/(1 - k^(2))`
D. `(d)kR`

Explanation

To find the maximum height of a projectile fired from the surface of the Earth with a velocity \( k v_e \) (where \( v_e \) is the escape velocity and \( k < 1 \)), we can use the principle of conservation of mechanical energy.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- The projectile is fired from the surface of the Earth with an initial velocity \( k v_e \).
- The escape velocity \( v_e \) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{2gR} \]
- Here, \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

2. **Calculate Initial Kinetic Energy (KE)**:
- The initial kinetic energy of the projectile can be expressed as:
\[ KE_i = \frac{1}{2} m (k v_e)^2 = \frac{1}{2} m (k^2 v_e^2) \]
- Substituting \( v_e^2 \):
\[ KE_i = \frac{1}{2} m (k^2 (2gR)) = mgR k^2 \]

3. **Calculate Initial Potential Energy (PE)**:
- The potential energy at the surface of the Earth is:
\[ PE_i = -\frac{GMm}{R} \]
- Here, \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

4. **Total Initial Energy (E_i)**:
- The total initial mechanical energy is the sum of kinetic and potential energy:
\[ E_i = KE_i + PE_i = mgR k^2 - \frac{GMm}{R} \]

5. **Final Conditions at Maximum Height**:
- At the maximum height \( h \), the velocity of the projectile becomes zero, so the kinetic energy is:
\[ KE_f = 0 \]
- The potential energy at height \( h \) is:
\[ PE_f = -\frac{GMm}{R + h} \]

6. **Total Final Energy (E_f)**:
- The total final mechanical energy is:
\[ E_f = KE_f + PE_f = 0 - \frac{GMm}{R + h} \]

7. **Apply Conservation of Energy**:
- According to the conservation of mechanical energy:
\[ E_i = E_f \]
- Therefore, we have:
\[ mgR k^2 - \frac{GMm}{R} = -\frac{GMm}{R + h} \]

8. **Rearranging the Equation**:
- Multiply through by \( R(R + h) \) to eliminate the fractions:
\[ mgR k^2 (R + h) - GMmR = -GMmR \]
- This simplifies to:
\[ mgR k^2 (R + h) = 0 \]

9. **Solving for Maximum Height \( h \)**:
- Rearranging gives:
\[ h = \frac{k^2 R}{1 - k^2} \]

10. **Total Height from the Center of the Earth**:
- The total height from the center of the Earth is:
\[ H = R + h = R + \frac{k^2 R}{1 - k^2} = \frac{R(1 - k^2 + k^2)}{1 - k^2} = \frac{R}{1 - k^2} \]

### Final Answer:
The maximum height from the center of the Earth is:
\[ H = \frac{R}{1 - k^2} \]