In this video, we will be going through a topic covered in classical mechanics in first year and planetary science in second year, that we found more difficult than it need be. Kepler's laws and its applications in physics. Kepler's laws describe the orbits of planets around the sun or stars around the galaxy in classical mechanics. They have been used to predict the orbits of many objects, such as asteroids and comets. They were also pivotal in the discovery of dark matter in the Milky Way. Johannes Kepler was born on December 27th, 1571 and was an important figure in the scientific revolution of the 17th century. He was a large supporter of the Copernican model of the solar system, and that's the sun being at the center of the universe with the planets orbiting it, otherwise known as Helio-Centric, but found a lot of difficulty in reconciling precise observations of Mars's orbit with the predicted orbit. This was the foundation of his early research, which resulted in his famous laws, the first two of which were published when working with the Danish astronomer Tyco Brahe in 1609 and the third being published 10 years later. Kepler's laws can be used to explore more sophisticated models of gravity and planets' remotion. Most problems related to the period and eccentricity of celestial orbits are still best solved using Kepler's laws, since they are much simpler to perform in calculations. Kepler's laws can also be applied to a model man-made objects like rockets and satellites in orbit. We want to look at Kepler's laws in more detail, see how they can be applied and show you some of the derivations for useful equations that are derived from these laws. The first law states that all planets move in elliptical orbits with a sun as one of the folky. Let's look at Kepler's second law, the lower-variate state that aligns that connects a planet to the sun, sweeps out equal areas at a constant rate. The angular momentum l, one object, for example, planet, orbiting a star, is l equals m v cross r, where m is the mass of the planet and v is the velocity of the planet and r is a vector pointing from the star to the planet. Under small angles, the pass can be uppercumulated as small straight line segments. We can then define a small change in the pass of the planet as the product of the momentum of the planet divided by its mass times a small time interval. Using the same approximation, we can obtain a formula for the error by trying to go by taking the cross product of r and delta r. This gives us the error as fed by the planet and run infinite as a small time interval. Placking the formula for the pass in and you are inching. Angle and momentum l is constant since the gravitational force acting on the planet is central. This means delta a over delta t is also constant, which is Kepler's second law. In other words, the second law of Kepler is a consequence of the conservation of angular momentum and the central force and is independent of the mass of the planet orbiting. Lastly, the law of period states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. In order to derive Kepler's third law, we must consider the eccentricity of ellipsis. Accenticity is a measure of how much a cornic section deviates from a circle. For ellipsis, eccentricity has a value between zero and one. We can insert the relation between the semi-major and semi-minor axis into the equation for the area swept out by the planet. This can be inserted into Kepler's second law and rearranged for the time period of the orbit. Using the relation between angular momentum and eccentricity, where m is the mass of the planet, k is equal to gmm, and a times 1 minus e squared is the semi-latus rectum. We can simplify this and cancel terms to obtain Kepler's third law.