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 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)

Answer 2

To determine the kinetic energy required to project a body from Earth's surface to infinity, we need to understand the concept of escape velocity and gravitational potential energy.

The escape velocity is the minimum speed needed for an object to escape from the gravitational influence of a celestial body without any further propulsion. The escape velocity (\(v_e\)) from the surface of the Earth is given by the formula:

\(v_e = \sqrt{2gR_E}\)

where:

• \(g\) is the acceleration due to gravity on Earth's surface.
• \(R_E\) is the radius of the Earth.

The kinetic energy (\(KE\)) required for the body to achieve this escape velocity is given by:

\(KE = \frac{1}{2}mv_e^2\)

Substituting the formula for escape velocity, we have:

\(KE = \frac{1}{2}m(2gR_E)\)

Simplifying further:

\(KE = mgR_E\)

Thus, the kinetic energy required to project a body from the Earth's surface to infinity is \(mgR_E\).

Let's evaluate the given options:

• \(2mgR_E\) - This is incorrect as it does not match our derived formula.
• \(mgR_E\) - This is correct based on the derivation above.
• \(\frac{1}{2}mgR_E\) - This is incorrect as it is less than the required kinetic energy.
• \(4mgR_E\) - This is incorrect as it exceeds the required kinetic energy.

Therefore, the correct answer is \(mgR_E\).