………….. is the minimum velocity with which a body must be projected vertically upwards in order to escape earth's gravitational field.

Explanation

To find the minimum velocity with which a body must be projected vertically upwards to escape Earth's gravitational field, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Escape Velocity
Escape velocity is the minimum velocity required for an object to break free from the gravitational attraction of a celestial body without any further propulsion.

### Step 2: Set Up the Energy Conservation Equation
When a body of mass \( m \) is projected upwards, it has both kinetic energy and gravitational potential energy. The total mechanical energy at the surface of the Earth can be expressed as:
\[
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