The most famous periodic comet is Halley's comet, which was the first periodic comet to be discovered.
a.Who discovered it and when did this occur? How was it shown that Halley's orbit is periodic?
Edmund Halley discovered this comet is periodic. He observed it in 1682, and published his idea that the comet was periodic in 1705. It was shown that the orbit is periodic using Newton's laws of motion.
b. Describe its orbit, include the period, ellipticity, and any other relevant information.
Period = 76 years (varies from about 75 to 79 years), ellipticity = 0.967, closest point to Sun = 0.586 AU, furthest point from the Sun = 35.1 AU, orbital inclination = 162.3 degrees (or 17.7 degrees)
c. To get a feeling for how fast Halley moves when it gets close to the Sun, compared to how slow it moves when it is far away, use Kepler's Second Law to estimate the ratio of the speed on its point of closest approach to its speed on the point fo furthest distance from the Sun.
Remeber that the velocity of the object is related to the distance from the central gravitational object (In this case, our Sun!) Kepler's Second law says that equal areas of the orbit are swept out in equal time. The easiest way to find the ratio of the velocities is just to take a ratio of the distances for aphelion and perihelion. 35/0.6 ~ 60
d. Is there a risk that Halley might collide with the Earth? If so, why, if not, why not??
No, Comet Halley is no danger to us (barring some freak gravitational interaction which is also very unlikely.) It turns out that Halley spends most of its time outside of the plane of the solar system. The points where its orbit intersects are between Venus and Earth, and between Mars and Jupiter.
Check out this link for a visual demonstration