A projectile is fired vertically upward 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 of finding the maximum height to which a projectile will rise when fired vertically upward 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 energy.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( v_e \) from the surface of the Earth is given by the formula:
\[
v_e = \sqrt{\frac{2GM}{R}}
...