Question

I need help completing the WHOLE problem, parts A, B, C, and D. I know it is a long problem, would appreciate labelled and clear steps, thank you.

Kepler's Laws I. A planet revolves around the sun in an elliptical orbit with the sun at one focus. 2. The line joining the sun to a planet sweeps out equal areas in equal times. 3. The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of its orbit. Kepler formulated these laws because they fitted the astronomical data. He wasn't able to see why they were true or how they related to each other. But Sir Isaac Newton, in his Principia Mathematica of 1687, showed how to deduce Kepler's three laws from two of Newton's own laws, the Second Law of Motion and the Law of Universal Gravitation. In Section 13.4 we proved Kepler's First Law using the calculus of vector functions. In this project we guide you through the proofs of Kepler's Second and Third Laws and explore some of their consequences 1. Use the following steps to prove Kepler's Sccond Law. The notation is the same as in the proof of the First Law in Section 13.4. In particular, use polar coordinates so that (a) Show that h-2k dt (b) Deduce that h (c) IfA- A() is the area swept out by the radius vector r rt) in the time interval [tn, dt as in the figure, show that dA dt do dt rin) (d) Deduce that -= th = constant dr This says that the rate at which A is swept out is constant and proves Kepler's Second Law

Answer 1

"Given r = (r cos theta) i + (r sin theta)j given position vector vector r = (r cos theta) i + (r sin theta)j then the velocity vector is vector V = r(r cos theta) i + (r sin theta)j + r theta (-sin theta + cos theta)j = rr + r theta theta vector v = (r cos theta) i + (r sin theta)j + (r sin theta + r theta cos theta) \r\n Now the angular momentum of a particle (with unit mass) vector h = vector r times vector v =