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.
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.
Show Hint
- A False but $\mathbf{R}$ is true
- Both $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$
- $\mathbf{A}$ is true but $\mathbf{R}$ is false
- Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$
The Correct Option is A
Solution and Explanation
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.
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