The kinetic energy needed to project a body of mass `m` from the earth surface (radius R) to infinity is

Options

A. `(mgR)/(2)`
B. `2mgR`
C. `mgR`
D. `(mgR)/(2)`

Explanation

To find the kinetic energy needed to project a body of mass \( m \) from the Earth's surface (radius \( R \)) to infinity, we can follow these steps:

### Step 1: Understand Escape Velocity
The escape velocity (\( v_e \)) is the minimum velocity required for an object to break free from the gravitational attraction of a celestial body without any further propulsion. The formula for escape velocity from the surface of the Earth is given by:

\[ v_e = \sqrt{2gR} \]

where:
- \( g \) is the acceleration due to gravity at the Earth's surface,
- \( R \) is the radius of the Earth.

### Step 2: Calculate Kinetic Energy
The kinetic energy (\( KE \)) required to project the body to infinity is given by the formula:

\[ KE = \frac{1}{2} mv_e^2 \]

Substituting the expression for escape velocity into this equation:

\[ KE = \frac{1}{2} m (\sqrt{2gR})^2 \]

### Step 3: Simplify the Expression
Now, simplify the expression:

\[ KE = \frac{1}{2} m (2gR) \]

This simplifies to:

\[ KE = mgR \]

### Final Answer
Thus, the kinetic energy needed to project a body of mass \( m \) from the Earth's surface to infinity is:

\[ KE = mgR \]