Prove the Kepler's first law
Answers
Deriving Kepler’s Laws from the Inverse-Square Law
Of course, Kepler’s Laws originated from observations of the solar system, but Newton’s great achievement was to establish that they follow mathematically from his Law of Universal Gravitation and his Laws of Motion. We present here a calculus-based derivation of Kepler’s Laws. You should be familiar with the results, but need not worry about the details of the derivation—it’s just here in case you’re curious.
The standard approach in analyzing planetary motion is to use (r, q ) coordinates, where r is the distance from the origin—which we take to be the center of the Sun—and q is the angle between the x-axis and the line from the origin to the point in question.
In the picture above, in which I have greatly exaggerated the ellipticity of the orbit, suppose the planet goes from A to B, a distance Ds, in a short time Dt, so its speed in orbit is Ds/Dt. Notice that the velocity can be resolved into vector components in the radial direction Dr/Dt (in this case negative) and in the direction perpendicular to the radius, rDq/Dt. The short line BC in the diagram above is perpendicular to SB (S being the center of the Sun), and therefore becoming perpendicular to SC as well in the limit of AB becoming an infinitesimally small distance.
In the limit of small Ds, then, we have where the angular velocity .
Answer:
Explanation:
Kepler's first law states that the planets travel around the sun in elliptical orbits, with the sun positioned at one of the ellipse's foci. ... He was forced to dispose of the idea of circular planetary orbits, and had to reject the ancient belief that the planets traveled their orbits with a consistent speed.
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