A projectile is fired vertically upwards from the surface of earth with a velocity `Kupsilon_(e)` where `upsilon_e` is the escape velocity and `K lt 1` Neglecting air resistance , show that the maximum height to which it will rise measured from the centre of earth is `R//(1-K^2)` where R is the radius of the earth .

Explanation

To solve the problem, we need to analyze the motion of a projectile fired vertically upward from the surface of the Earth with a velocity \( K \upsilon_e \), where \( \upsilon_e \) is the escape velocity and \( K < 1 \). We will use the principle of conservation of energy to find the maximum height reached by the projectile.

### Step-by-Step Solution:

1. **Identify the Escape Velocity**:
The escape velocity \( \upsilon_e \) is given by the formula:
\[
\upsilon_e = \sqrt{\frac{2GM}{R}}
...