A particle is projected vertically upwards the surface of the earth (radius `R_(e))` with a speed equal to one fourth of escape velocity what is the maximum height attained by it from the surface of the earth?

Explanation

To solve the problem of finding the maximum height attained by a particle projected vertically upwards with a speed equal to one fourth of the escape velocity, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand the Escape Velocity
The escape velocity \( v_e \) from the surface of the Earth can be expressed as:
\[ v_e = \sqrt{\frac{2GM}{R_e}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, and \( R_e \) is the radius of the Earth.
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