申論題

(4) (30 points)

(a) (10 points) In a polar coordinate system, the unit vector in the direction of r is and in the direction of Φ is . The coordinates of a particle in this coordinate is (r(t), Φ(t)). What is its position vector of the particle? Work out the velocity and acceleration of the particle in the directions
 The Sun is at the origin of a polar coordinate system. A planet is orbiting the Sun in elliptical orbit. Its distance from the Sun is r. The angle between its position vector and the semi-major axis of the orbit is Φ.
The three Kepler's laws can be expressed as:
 First law:

where P is the orbital period, e is the eccentricity, a is the semi-major axis, and b =

is the semi-minor axis.