A particle is projected vertivally upwards from the surface of earth `(`radius` R_e)` with a kinetic energy equal to half of the minimum value needed for it to escape. The height to which it rises above the surface of earth is ……

Explanation

To solve the problem, we need to determine the height to which a particle rises when it is projected vertically upwards from the surface of the Earth with a kinetic energy equal to half of the minimum value needed for it to escape Earth's gravitational pull.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The minimum kinetic energy required to escape the gravitational field of the Earth can be derived from the escape velocity formula:
\[
v_e = \sqrt{\frac{2GM}{R_e}}
...