[M] According to Kepler’s first law, a comet should have

Chapter 6, Problem 11E

(choose chapter or problem)

[M] According to Kepler’s first law, a comet should have an elliptic, parabolic, or hyperbolic orbit (with gravitational attractions from the planets ignored). In suitable polar coordinates, the position \((r, \vartheta)\) of a comet satisfies an equation of the form

\(r=\beta+e(r \cdot \cos \vartheta)\)

where \(\beta\) is a constant and e is the eccentricity of the orbit, with \(0 \leq e<1\) for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola. Suppose observations of a newly discovered comet provide the data below. Determine the type of orbit and predict where the comet will be when \(\vartheta=4.6\) (radians).\(^3\)