JEE Main 2025 — Gravitation Question with Solution

Let's analyze both the Assertion and the Reason step by step.

Calculation of the Required Kinetic Energy:

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

$$U(r)=-\frac{GMm}{r},$$

where $$G$$ is the gravitational constant and $$M$$ is the mass of the Earth.

At the Earth's surface (where $$r=R$$), the potential energy is:

$$U(R)=-\frac{GMm}{R}.$$

At infinity, the potential energy is defined as:

$$U(\infty )=0.$$

The energy needed to move the body from the Earth’s surface to infinity is the change in potential energy:

$$\Delta U=U(\infty )-U(R)=0-(-\frac{GMm}{R})=\frac{GMm}{R}.$$

Using the relation $$g=\frac{GM}{R^2},$$ we find:

$$\frac{GMm}{R}=mgR.$$

So, the required kinetic energy to just reach infinity (with zero speed at infinity) is:

$$\text{Kinetic Energy}=mgR.$$

The Assertion A states that the required kinetic energy is $$\frac{1}{2}mgR$$, which is only half of the actual value.

Verification of the Reason:

The Reason states: "The maximum potential energy of a body is zero when it is projected to infinity from earth surface."

By convention in gravitational problems, the potential energy at infinity is taken to be zero. This is a standard and correct statement.

Conclusion:

Assertion A is false because it underestimates the needed energy; the correct kinetic energy should be $$mgR$$.

Reason R is true since the gravitational potential energy is indeed defined to be zero at infinity.

Therefore, the correct answer is:

Option A: $$\mathbf{A}$$ is false but $$\mathbf{R}$$ is true.