Question

To project a body of mass \( m \) from Earth's surface to infinity, the required kinetic energy is (assume, the radius of Earth is \( R_E \), \( g \) = acceleration due to gravity on the surface of Earth):

• A. \( 2mgR_E \)
• B. \( mgR_E \)
• C. \( \frac{1}{2}mgR_E \)
• D. \( 4mgR_E \)

Answer 1

The Correct Option is B

The total energy needed to project an object to infinity equals the work required to counteract the gravitational potential energy at Earth's surface.

Gravitational potential energy at Earth's surface:

\[ \text{Potential Energy} = -\frac{GMm}{R_E}, \]

Here, \( G \) represents the gravitational constant, \( M \) is Earth's mass, \( m \) is the object's mass, and \( R_E \) is Earth's radius.

Kinetic energy needed for escape velocity:

\[ K = \frac{1}{2}mv_e^2. \]

Equating this to the energy required to overcome gravitational pull:

\[ \frac{1}{2}mv_e^2 = \frac{GMm}{R_E}. \]

Substituting \( g = \frac{GM}{R_E^2} \):

\[ \frac{GMm}{R_E} = mgR_E. \]

Therefore, the required kinetic energy is:

\[ K = mgR_E. \]

Final Answer: \( mgR_E \) (Option 2)