A mass `m` is taken to a height `R` from the surface of the earth and then is given a vertical velocity `upsilon`. Find the minimum value of `upsilon`, so that mass never returns to the surface of the earth. (Radius of earth is `R` and mass of the earth `m`).

Explanation

To find the minimum value of the vertical velocity \( \upsilon \) that a mass \( m \) must have when taken to a height \( R \) above the Earth's surface so that it never returns to the surface, we can follow these steps:

### Step 1: Understand the Energy Conservation Principle
The total mechanical energy (kinetic energy + potential energy) of the mass must be conserved. We will analyze the energy at the height \( R \) above the Earth's surface and at infinity.

### Step 2: Calculate Initial Total Energy
At a height \( R \) from the surface of the Earth, the total initial energy \( E_i \) consists of:
- Kinetic Energy \( KE_i = \frac{1}{2} m \upsilon^2 \)
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