Consider the following statements.
Kepler’s second law is based on the law of conservation of angular momentum.
Kepler’s third law is based on Newton’s law of gravitation.

Options

A. Only I Is correct
B. Only II Is correct
C. Both the statements are correct
D. None Is correct

Answer

To analyze the statements regarding Kepler's laws, let's break down the information step by step.

### Step 1: Understanding Kepler's Second Law
Kepler's Second Law, also known as the Law of Equal Areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that the planet moves faster when it is closer to the Sun and slower when it is farther away.

**Hint for Step 1:** Recall that the area swept out by the planet in a given time interval is constant, which relates to angular momentum.

### Step 2: Relation to Angular Momentum
The conservation of angular momentum states that if no external torque acts on a system, the angular momentum of that system remains constant. In the case of a planet orbiting the Sun, the gravitational force acts as a central force, which does not exert torque about the center of the orbit (the Sun). Thus, the angular momentum of the planet is conserved.

**Hint for Step 2:** Consider how the distance from the Sun affects the speed of the planet and how this relates to the area swept out.

### Step 3: Understanding Kepler's Third Law
Kepler's Third Law states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit. Mathematically, this is expressed as \( T^2 \propto r^3 \).

**Hint for Step 3:** Think about how this law relates to the gravitational forces acting on the planets and their distances from the Sun.

### Step 4: Relation to Newton's Law of Gravitation
Newton's law of gravitation states that the force of attraction between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Kepler's Third Law can be derived from Newton's law of gravitation, as it relates the gravitational force to the motion of the planets.

**Hint for Step 4:** Consider how the gravitational force influences the orbital motion and leads to the relationship described in Kepler's Third Law.

### Step 5: Evaluating the Statements
- **Statement 1:** Kepler’s second law is based on the law of conservation of angular momentum. (This statement is **True**.)
- **Statement 2:** Kepler’s third law is based on Newton’s law of gravitation. (This statement is **True**.)

### Conclusion
Both statements are correct. Therefore, the correct answer is that both statements are true.

**Final Answer:** Option 3: Both statements are correct.