Let's analyze the problem and establish the correct relationships using physics concepts, specifically in the context of celestial mechanics and gravitational physics.
We start by identifying each term given in List-I and aligning it with the correct expression from List-II:
Kinetic Energy of Earth(A): The kinetic energy of a body in orbit can be given by the formula:
T = \frac{1}{2}\frac{GM_sM_e}{a}
This matches with option (II) \frac{GM_sM_e}{2a}.
Potential Energy of Earth and Sun(B): The gravitational potential energy between two masses is given by:
U = -\frac{GM_sM_e}{a}
This directly matches with the expression in (I) -\frac{GM_sM_e}{a}.
Total Energy of Earth and Sun(C): The total mechanical energy of an orbiting system is:
E = T + U = -\frac{GM_sM_e}{2a}
This corresponds with (IV) -\frac{GM_sM_e}{2a}.
Escape Energy from Surface of Earth per unit mass(D): The escape velocity energy per unit mass from a celestial body is described by:
E_{\text{escape}} = \frac{GM_e}{R}
This relationship corresponds with (III) \frac{GM_e}{R}.
By matching the expressions with the correct terms, we arrive at the pairing:
A → 2, B → 1, C → 4, D → 3
Thus, the correct choice is A → 2, B →1, C → 4, D → 3.
For deeper insight, remember these concepts:
Gravitational potential energy is typically negative because it's calculated with the assumption that the force is attractive.
In a stable orbit, the kinetic energy is determined by balancing gravitational attraction with rotational motion.
The escape energy is the energy needed per unit mass to leave a celestial body's gravitational influence.
