Match List I with List II :

(Where $$a$$ = radius of planet orbit, $$r$$ = radius of planet, $$M$$ = mass of Sun, $$m$$ = mass of planet)

For a planet revolving around the sun in a circular orbit of radius aaa:

The gravitational force provides the centripetal force.

$$\frac{GMm}{a^2}=\frac{mv^2}{a}$$

From this, we get the orbital velocity:

$$v^2=\frac{GM}{a}$$

Now kinetic energy of the planet is:

$$K=\frac{1}{2}mv^2=\frac{1}{2}m\cdot\frac{GM}{a}=\frac{GMm}{2a}$$

So, (A) matches with (II)

Gravitational potential energy of the sun-planet system is:

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

So, (B) matches with (I)

Total mechanical energy is:

$$E=K+U=\frac{GMm}{2a}-\frac{GMm}{a}=-\frac{GMm}{2a}​$$​

So, (C) matches with (IV)

Escape energy per unit mass at the surface of a planet is:

$$\ \frac{\ 1}{2}​v_e^2​=\ \frac{\ GM}{r}$$

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

Energy per unit mass:

$$v_e^2=\frac{GM}{r}$$

So, (D) matches with (III)

Thus, the correct matching is:

(A)−(II), (B)−(I), (C)−(IV), (D)−(III)