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

Answer: (B)

$mgR_E$

Answer 2

The total energy required to project a body to infinity is equal to the work needed to overcome the gravitational potential energy at the surface of the Earth.

Gravitational potential energy at the surface of Earth:

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

where $G$ is the gravitational constant, $M$ is the mass of the Earth, $m$ is the mass of the body, and $R_E$ is the radius of the Earth.

The kinetic energy required for escape velocity:

$K = \frac{1}{2}mv_e^2.$

Equating this to the energy needed to overcome the gravitational pull:

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

Substituting $g = \frac{GM}{R_E^2}$, we get:

$\frac{GMm}{R_E} = mgR_E.$

Thus, the required kinetic energy is:

$K = mgR_E.$

Final Answer: $mgR_E$ (Option 2)