Sec1Sess21

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Kepler's Second Law
Motion of planets is in a plane,
and the area is swept out (encompased inside) by the line from
sun to planet at a constant rate.

once orbit is known than it can be found how fast
a planet moves on that orbit

Newton
Later explained the 2cnd law by the formula for
gravitational attraction

Kepler's law in terms of vectors?

express area in terms of vectors
eclipse vectors are 1. center to starting point and 2. center to point traveled to
area approximately equals 1/2 area created by parallelogram of vectors in elipse
<math>\mathrm{\approx \ 1/2|}\vec{r}\mathrm{\ x\ change(}\vec{r}\mathrm{)|\ where\ r}</math> is the eclipse position vectors notation
= area swept after time occurs change(time) (small)
<math>\mathrm{change}(\vec{r})\ \approx \ \vec{v}\ *\ \mathrm{change(t)\ therefore}</math>
<math>\mathrm{1/2*|}\vec{r}\ \mathrm{x\ change}(\vec{r}\mathrm{)|\ \approx \ 1/2\ *\ |}\vec{r}\mathrm{\ x\ }\vec{v}\mathrm{|\ *\ change(t)}</math>
change(t) is able to be taken out of length section because it is a scalar

law says: |<math>\vec{r}\ \mathrm{x}\ \vec{v}</math>| = constant = rate at which area is swept by position vector

Part of law 2:
plane of motion
both eclipse vectors are in the plane
vector1: center to starting point
vector2: velocity of point moving on eclipse

Direction of <math>\vec{r}</math> x -> v is normal to the plane of motion
(motion will stay in the plane)

kepler's 2cnd law <=> <math>\vec{r}</math> x -> v = constant vector
<math>\mathrm{<=>\ d(}\vec{r}\mathrm{\ x\ }\vec{v}\mathrm{) = 0}</math>

Product rule is ok for taking derivative of dot product and cross product
Order of vectors matters at least for cross product.
similar to product rule: d(r . s) = d(r) . s + r . d(s)
d(r x s) = d(r) x s + r x d(s)

d(r)/d(t) x v + r x d(v)/d(t) = 0
v = d(r)/d(t)
a = d(v)/d(t)
v x v + r x a = 0

a vector cross itself is always 0
r x a = 0 because the acceleration is parallel to the position
this is stated in keplar's 2cnd law
<=> gravitational force is parallel to the position vector

For motion under any type of central force, the path of motion will lie in a plane and area will be swept out by the radius vector at a constant rate