other mass takes place in a plane. Construct a set of x, y axes with origin located at
the center of mass of M. Further, let eˆr= cos θˆe 1 + sin θˆe 2 denote a unit vector at the
origin of our coordinate system and pointing in the direction of the mass m. One can
then express the vector force of attraction of mass Mon mass mby the equation
F=−GmM
r^2ˆer (9 .102)To find the equation of motion of mass mwith respect to mass M, use Newton’s
second law. Let r =rˆer denote the position vector of mass mwith respect to our
origin. The equation of motion of mass mis determined from Newton’s second law
and is
F =−GmM
r^2 eˆr=md^2 r
dt^2 =mdV
dt (9 .103)From this equation it can be shown that the motion of mass mcan be described as
a conic section. In order to accomplish this, let us review some facts about conic
sections.
Recall that a conic section was defined as a locus of points P(x, y )such that the
distance of P from a fixed point (or points), called a focus, is proportional to the
distance of P from a fixed line, called the directrix. The constant of proportionality
is called the eccentricity and is denoted by the symbol
. If = 1, a parabola results;