Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: The kinetic energy needed to project a body of mass $m$ from earth surface to infinity is $\frac{1}{2} \mathrm{mgR}$, where R is the radius of earth. Reason R: The maximum potential energy of a body is zero when it is projected to infinity from earth surface.

• A. A False but $\mathbf{R}$ is true
• B. Both $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$
• C. $\mathbf{A}$ is true but $\mathbf{R}$ is false
• D. Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$

Hint:

The kinetic energy needed to project a body to infinity is equal to the potential energy at the earth's surface.

Answer 1

The Correct Option is A

Assertion (A): The kinetic energy needed to project a body of mass \( m \) from the Earth’s surface to infinity is \( \frac{1}{2} mgR \), where \( R \) is the radius of the Earth.

Reason (R): The maximum potential energy of a body is zero when it is projected to infinity from the Earth’s surface.

Concept Used:

The gravitational potential energy of a body of mass \( m \) at a distance \( r \) from the center of the Earth is given by:

\[ U = -\frac{GMm}{r} \]

where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. At infinity, \( U = 0 \). The minimum kinetic energy required to just escape from Earth's gravitational field is known as the escape energy.

It is given by:

\[ K = \frac{1}{2} m v_e^2 \] \[ \text{where } v_e = \sqrt{\frac{2GM}{R}} \] \[ \therefore K = \frac{GMm}{R} \]

Step-by-Step Solution:

Step 1: Compute the correct expression for the required kinetic energy.

\[ K = \frac{GMm}{R} \]

Step 2: Relate \( \frac{GM}{R^2} \) to \( g \), the acceleration due to gravity.

\[ g = \frac{GM}{R^2} \Rightarrow GM = gR^2 \] \[ K = \frac{gR^2 m}{R} = mgR \]

Step 3: Therefore, the kinetic energy needed to project a body from the Earth's surface to infinity is:

\[ K = mgR \]

This shows that the Assertion (A) is false because it states \( \frac{1}{2} mgR \) instead of \( mgR \).

Step 4: Analyze the Reason (R).

The Reason correctly states that the maximum potential energy (at infinity) is zero, since gravitational potential energy becomes zero at infinite separation.

Final Computation & Result:

• Assertion (A): False
• Reason (R): True

Final Answer: Assertion (A) is false, but Reason (R) is true.