Question

Match list-I with list-II:

List-I	List-II
(A) Kinetic energy of planet	\(- \frac{GMm}{a}\)
(B) Gravitational Potential energy of Sun-planet system	\(- \frac{GMm}{2a}\)
(C) Total mechanical energy of planet	\(\frac{GM}{r}\)
(D) Escape energy at the surface of planet for unit mass object	\(- \frac{GMm}{2a}\)

(Where a = radius of planet orbit, r = radius of planet, M = mass of Sun, m = mass of planet) Choose the correct answer from the options given below:

• (A) – II, (B) – I, (C) – IV, (D) – III
• (A) – III, (B) – IV, (C) – I, (D) – II
• (A) – I, (B) – IV, (C) – II, (D) – III
• (A) – I, (B) – II, (C) – III, (D) – IV

Answer 1

The Correct Option is A

The kinetic energy (KE) of a planet is given by:

\[ \text{KE} = \frac{1}{2} mv^2 = \frac{GMm}{2a} \]

The gravitational potential energy (PE) of the Sun-planet system is:

\[ \text{PE} = -\frac{GMm}{a} \]

The total mechanical energy (TE) of the planet is:

\[ \text{TE} = \text{KE} + \text{PE} = -\frac{GMm}{2a} \]

Escape energy at the surface of the planet for a unit mass object is given by:

\[ \text{Escape Energy} = \frac{Gm}{r} \]

Answer 2

Step 1: Recall the relevant formulas 
For a planet of mass \( m \) orbiting the Sun of mass \( M \) at a distance \( a \):

• Kinetic energy of planet, \( K = \dfrac{GMm}{2a} \)
• Gravitational potential energy of the Sun–planet system, \( U = -\dfrac{GMm}{a} \)
• Total mechanical energy of planet, \( E = K + U = -\dfrac{GMm}{2a} \)
• Escape energy per unit mass from surface of planet, \( \varepsilon = \dfrac{GM}{r} \)

Step 2: Match each quantity correctly
 

List-I	List-II
(A) Kinetic energy of planet	\( +\dfrac{GMm}{2a} \)
(B) Gravitational potential energy of Sun–planet system	\( -\dfrac{GMm}{a} \)
(C) Total mechanical energy of planet	\( -\dfrac{GMm}{2a} \)
(D) Escape energy at the surface of planet for unit mass object	\( +\dfrac{GM}{r} \)

Step 3: Final correspondence
\[ (A) - II, \quad (B) - I, \quad (C) - IV, \quad (D) - III \]

Final answer
 

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