With what velocity must a body be thrown from earth's surface so that it may reach a maximum height of `4R_(e)` above the Earth's surface ? (Radius of the Earth `R_(e)=6400 km,g=9.8 ms^(-2)`).

Explanation

To find the velocity with which a body must be thrown from the Earth's surface to reach a maximum height of \( 4R_e \) (where \( R_e = 6400 \, \text{km} \)), we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to determine the initial velocity \( V \) required for a body to reach a height of \( 4R_e \) above the Earth's surface. At this maximum height, the velocity of the body will be zero.

### Step 2: Define the Energies
At the Earth's surface (point 1), the total mechanical energy consists of kinetic energy (KE) and gravitational potential energy (PE):
- Kinetic Energy at point 1:
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