The kinetic energy needed to project a body of mass m from the earth's surface to infinity is

Options

A. `1/4mgR`
B. `1/2mgR`
C. `mgR`
D. `2mgR`

Answer

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

### Step 1: Understand the concept of escape velocity
Escape velocity is the minimum velocity needed for an object to break free from the gravitational attraction of a celestial body without any further propulsion. For Earth, this velocity can be derived from the balance of kinetic and gravitational potential energy.

### Step 2: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

### Step 3: Write the formula for escape velocity
The escape velocity \( v_{\text{escape}} \) from the Earth's surface can be derived from the gravitational potential energy and is given by:
\[ v_{\text{escape}} = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)) and \( R \) is the radius of the Earth.

### Step 4: Substitute escape velocity into the kinetic energy formula
Substituting \( v_{\text{escape}} \) into the kinetic energy formula, we get:
\[ KE = \frac{1}{2} m (v_{\text{escape}})^2 = \frac{1}{2} m (2gR) \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ KE = \frac{1}{2} m (2gR) = mgR \]

### Conclusion
Thus, the kinetic energy needed to project a body of mass \( m \) from the Earth's surface to infinity is:
\[ KE = mgR \]

### Final Answer
The correct answer is \( mgR \).

---