A rocket of mass M is launched vertically from the surface of the earth with an initial speed `V= sqrt((gR)/(2))` . Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is `(R)/(X)` where X is:

Explanation

To solve the problem, we will use the principle of conservation of mechanical energy. The total mechanical energy of the rocket at launch will be equal to the total mechanical energy at the maximum height it reaches.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The rocket is launched from the surface of the Earth with an initial speed \( V = \sqrt{\frac{gR}{2}} \).
- The mass of the rocket is \( M \).
- The radius of the Earth is \( R \).
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