Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason $\mathbf{R}$
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.
In the light of the above statements, choose the correct answer from the options given below
Solution
<p>Let's analyze both the Assertion and the Reason step by step.</p>
<p>Calculation of the Required Kinetic Energy:</p>
<p><p>The gravitational potential energy of a body of mass $m$ at a distance $r$ from the center of the Earth is given by: </p>
<p>$U(r) = -\frac{GMm}{r},$ </p>
<p>where $G$ is the gravitational constant and $M$ is the mass of the Earth.</p></p>
<p><p>At the Earth's surface (where $r = R$), the potential energy is: </p>
<p>$U(R) = -\frac{GMm}{R}.$</p></p>
<p><p>At infinity, the potential energy is defined as: </p>
<p>$U(\infty) = 0.$</p></p>
<p><p>The energy needed to move the body from the Earth’s surface to infinity is the change in potential energy: </p>
<p>$\Delta U = U(\infty) - U(R) = 0 - \Bigg(-\frac{GMm}{R}\Bigg) = \frac{GMm}{R}.$</p></p>
<p><p>Using the relation $g = \frac{GM}{R^2},$ we find: </p>
<p>$\frac{GMm}{R} = mgR.$</p></p>
<p><p>So, the required kinetic energy to just reach infinity (with zero speed at infinity) is: </p>
<p>$\text{Kinetic Energy} = mgR.$</p></p>
<p><p>The Assertion A states that the required kinetic energy is $\frac{1}{2}mgR$, which is only half of the actual value.</p></p>
<p>Verification of the Reason:</p>
<p><p>The Reason states: "The maximum potential energy of a body is zero when it is projected to infinity from earth surface."</p></p>
<p><p>By convention in gravitational problems, the potential energy at infinity is taken to be zero. This is a standard and correct statement.</p></p>
<p>Conclusion:</p>
<p><p>Assertion A is false because it underestimates the needed energy; the correct kinetic energy should be $mgR$.</p></p>
<p><p>Reason R is true since the gravitational potential energy is indeed defined to be zero at infinity.</p></p>
<p>Therefore, the correct answer is:</p>
<p>Option A: $\mathbf{A}$ is false but $\mathbf{R}$ is true.</p>
About this question
Subject: Physics · Chapter: Gravitation · Topic: Kepler's Laws
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