JEE Physics 2025 Question with Solution

Given: Assertion says the kinetic energy needed to project a body of mass $m$ from the Earth's surface to infinity is $\frac{1}{2}mgR$. Reason says the maximum potential energy of a body is zero at infinity.

Find: Which option correctly describes the truth of Assertion A and Reason R.

The gravitational potential energy at distance $r$ from the centre of the Earth is

$$U=-\frac{GMm}{r}$$

At infinity, gravitational potential energy becomes

$$U=0$$

The minimum kinetic energy required to take the body from the Earth's surface to infinity is the escape energy:

$$K=\frac{1}{2}mv_e^2$$

where

$$v_e=\sqrt{\frac{2GM}{R}}$$

Therefore,

$$K=\frac{GMm}{R}$$

Using

$$g=\frac{GM}{R^2}$$

we get

$$GM=gR^2$$

Hence,

$$K=\frac{gR^2m}{R}=mgR$$

So the required kinetic energy is $mgR$, not $\frac{1}{2}mgR$. Therefore, Assertion A is false.

Now examine Reason R. Since gravitational potential energy is taken as zero at infinity, the statement that the maximum potential energy is zero at infinity is true.

Therefore, the correct conclusion is: Assertion A is false, but Reason R is true. The correct option is A.

Given: A body starts from the Earth's surface and is projected to infinity.

Find: Whether the assertion value of kinetic energy is correct.

At the Earth's surface, the gravitational potential energy is

$$U_s=-\frac{GMm}{R}$$

At infinity,

$$U_\infty=0$$

So the increase in potential energy needed is

$$\Delta U=U_\infty-U_s=0-\left(-\frac{GMm}{R}\right)=\frac{GMm}{R}$$

For the body to just reach infinity with zero final speed, the initial kinetic energy must equal this increase in potential energy:

$$K=\Delta U=\frac{GMm}{R}=mgR$$

Thus the assertion value $\frac{1}{2}mgR$ is incorrect.

The reason is true because gravitational potential energy is indeed zero at infinity, but the assertion fails because the numerical value of the required kinetic energy has been written incorrectly.

Hence, Assertion A is false and Reason R is true, so the correct option is A.