I was doing Kepler's first law (basically that is the only law in which I have a problem) other two laws are easy to understand). During the explanation of the first law my instructor said that we need to establish the fact that angular momentum does not change per unit time i.e. stays constant.
Problem 1.) Instructor says: "First of all, it can be shown that the angular momentum $L$ is constant. Since in the first term, the centripetal force $F$ is collinear with $r$, and in the second term $v$ is obviously collinear with $v$, both terms are zero, which implies that $L$ is constant and that the orbit lies in a plane!" What does this mean (specifically what does "lies in a plane mean"?)
Problem 2.) I learned Kepler's second law which was about equal areas and we PROVED that law of areas mathematically and also in his third law we PROVED, mathematically, that the period squared is equal to semi-major axis cubed, but in his first law which is:"All planets move about the Sun in elliptical orbits, having the Sun as one of the foci" we did not PROVE this law. We just said that the orbit lies in a plane, not that it is elliptical which is actually the law. Should not we be proving that the orbit is elliptical with Sun as on of the foci? Or am I missing something?
Problem 3.) Here is a screenshot from the website:
It says dr/dt=v (vector) and it is in the same direction as p (vector), but the change in r (vector) with respect to time is not v, it is the one component of v which is directed inwards. Obviously v is collinear with v but it is not "v" which is causing change in r, it is the component of v which causes the change in r and it is directed inwards, not collinear.