State law of orbit?

Plz help me as far as possible plz plz help

http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html

The state law of orbit, also known as Kepler's first law of planetary motion, states that all planets move in elliptical orbits around the sun, with the sun at one of the two foci of the ellipse. This law was developed by the German astronomer Johannes Kepler in the early 17th century.

But let me explain how to derive this law using the concept of gravitational force.

1. Start with Newton's law of universal gravitation, which states that any two objects in the universe attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This force is given by F = (G * m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

2. Consider a planet moving around the sun. The sun exerts a gravitational force on the planet, pulling it toward the sun. The planet also exerts a gravitational force on the sun, but since the sun is much more massive, it is considered stationary for our purposes.

3. The gravitational force acting on the planet provides the centripetal force required to keep the planet in orbit. The centripetal force is always directed toward the center of the circular or elliptical path.

4. Equating the gravitational force and the centripetal force, we get (G * m1 * m2) / r^2 = (m * v^2) / r, where m is the mass of the planet and v is its orbital velocity.

5. Simplifying the equation, we find v^2 = (G * M) / r, where M is the mass of the sun.

6. We can rewrite this equation as v^2 = GM / r, where G and M are constants.

7. Since the right-hand side of the equation is constant, this implies that the orbital velocity of the planet remains constant at any point in its orbit.

8. Now, let's consider an elliptical orbit. In an ellipse, the distance between the planet and the sun varies at different points in the orbit. The planet moves faster when it is closer to the sun and slower when it is farther away.

9. This variation in speed ensures that the planet sweeps out equal areas in equal times. In other words, the planet moves faster in the shorter arc of the ellipse and slower in the longer arc, so that it spends equal amounts of time in each area.

Therefore, Kepler's first law of planetary motion states that all planets move in elliptical orbits with the sun at one of the foci. This law can be derived from Newton's law of universal gravitation and the concept of centripetal force.