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Speed of Halley's Comet

CLMECH-NSH@XY

Halley’s comet orbits the sun in a highly elongated elliptical orbit. At its farthest point, it is 35.1 AU from the sun (AU = astronomical unit, the mean radius of Earth’s orbit, $1.496\times 10^{11}\,m$). From there, the sun’s gravitational field pulls it back into its closest point of 0.586 AU.

The force exerted by the sun’s gravity is:

$$F =\frac{GMm}{r^2}$$

...where $G$ is the universal gravitational constant, $6.674\times 10^{-11}\;m^3/kg\,s^2$; $M$ is the mass of the sun, $1.989\times 10^{30}\,kg$; $m$ is the mass of the comet; and $r$ is the distance from sun to comet.

Estimate the comet’s maximum speed by finding the work done by gravity as it falls from its farthest point to its closest. For this estimate, you can ignore any initial velocity the comet has at its farthest point.