A particle is projected vertically upwards with a velocity `sqrt(gR)`, where `R` denotes the radius of the earth and `g` the acceleration due to gravity on the surface of the earth. Then the maximum height ascended by the particle is

Answer

To find the maximum height ascended by a particle projected vertically upwards with an initial velocity of \( \sqrt{gR} \), where \( R \) is the radius of the Earth and \( g \) is the acceleration due to gravity, we can use the principle of conservation of mechanical energy.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The particle is projected with an initial velocity \( u = \sqrt{gR} \).
- The initial height \( h_0 = 0 \) (on the surface of the Earth).

2. **Calculate Initial Kinetic Energy (KE)**:
\[ KE_i = \frac{1}{2} m u^2 = \frac{1}{2} m (\sqrt{gR})^2 = \frac{1}{2} m gR \]

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

4. **At Maximum Height**:
- The final kinetic energy \( KE_f = 0 \) (the particle momentarily stops).
- The final potential energy at height \( h \) is:
\[ PE_f = -\frac{GMm}{R + h} \]

5. **Apply Conservation of Energy**:
The total mechanical energy at the initial point equals the total mechanical energy at the maximum height:
\[ KE_i + PE_i = KE_f + PE_f \]
Substituting the values:
\[ \frac{1}{2} m gR - \frac{GMm}{R} = 0 - \frac{GMm}{R + h} \]

6. **Simplify the Equation**:
Cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{2} gR - \frac{GM}{R} = -\frac{GM}{R + h} \]
Rearranging gives:
\[ \frac{1}{2} gR = \frac{GM}{R} - \frac{GM}{R + h} \]

7. **Substitute \( g \)**:
Recall that \( g = \frac{GM}{R^2} \):
\[ \frac{1}{2} \left(\frac{GM}{R^2}\right) R = \frac{GM}{R} - \frac{GM}{R + h} \]
Simplifying gives:
\[ \frac{GM}{2R} = \frac{GM}{R} - \frac{GM}{R + h} \]

8. **Cross-Multiply**:
\[ \frac{GM(R + h) - 2GM}{2R(R + h)} = 0 \]
This simplifies to:
\[ R + h - 2 = 0 \implies h = 2 - R \]

9. **Final Height Calculation**:
Rearranging gives:
\[ h = R \]

Thus, the maximum height ascended by the particle is equal to the radius of the Earth, \( h = R \).

### Final Answer:
\[ \text{Maximum height } h = R \]