Vector-valued functions provide a useful method for studying various curves both in the plane and in three-dimensional space. We can apply this concept to calculate the velocity, acceleration, arc length, and curvature of an object\u2019s trajectory. In this module, we examine these methods and show how they are used.<\/p>","rendered":"
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<\/p>\nFigure 1. Halley\u2019s Comet appeared in view of Earth in 1986 and will appear again in 2061.<\/p>\n<\/div>\n
In 1705, using Sir Isaac Newton\u2019s new laws of motion, the astronomer Edmond Halley made a prediction. He stated that comets that had appeared in 1531, 1607, and 1682 were actually the same comet and that it would reappear in 1758. Halley was proved to be correct, although he did not live to see it. However, the comet was later named in his honor.<\/p>\n
Halley\u2019s Comet follows an elliptical path through the solar system, with the Sun appearing at one focus of the ellipse. This motion is predicted by Johannes Kepler\u2019s first law of planetary motion, which we mentioned briefly in the\u00a0Module 1: Parametric Equations and Polar Coordinates<\/em>. In\u00a0the section\u00a0Motion in Space<\/em>, we show how to use Kepler\u2019s third law of planetary motion along with the calculus of vector-valued functions to find the average distance of Halley\u2019s Comet from the Sun.<\/p>\n Vector-valued functions provide a useful method for studying various curves both in the plane and in three-dimensional space. We can apply this concept to calculate the velocity, acceleration, arc length, and curvature of an object\u2019s trajectory. In this module, we examine these methods and show how they are used.<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t