With others plotting to kill the Holy Roman Emperor and overthrow the government, Johann Kepler (1571-1630) fled from his job as royal astrologer, astronomer, and mathematician in Prague. His understanding of the sky allowed him to sell horoscopes to provide subsistence for his family. While teaching school in Linz, Kepler attempted to improve his explanation of the planets' spacing by considering regular polygons that could be easily constructed. Guided by the Pythagorean penchant for harmonies, Kepler wrote music for each planet and considered the cause of their motion. Kepler suspected the rotating Sun equally swept the planets around the sky against a resistance proportional to the square root of the orbit radius, R. With a particular planet's period of orbit proportional to both the size of the orbit (T ∝ R) and the resistance, (T ∝ resistance ∝ √R), Kepler proposed that the period, T ∝ R3/2. This fits the actual periods of the planets!Kepler had now explained
Isaac Newton (1642-1727) an English introvert explained the needed cause. While his teaching job at Cambridge University was suspended due to the plague, he avoided infection by living on the family farm. Here he pondered the motion of the Moon and considered a possible relationship with falling bodies on Earth. Newton realized the orbit of the Moon could also be viewed as perpetually falling towards Earth but at a lesser rate perhaps due to reduced gravity further from Earth. Newton found his hunch that gravity faded with the square of the distance was supported by Kepler's laws describing the planets as they sampled gravity at different distances from the Sun. But the reclusive Newton kept his thoughts to himself.
Later, back teaching at Cambridge, Newton did experiments with prisms and wrote a controversial Theory of Light and Colors. Its publication in 1672, brought him recognition as a scientist and member of the Royal Society. In 1684 another member of the Royal Society, Edmond Halley, asked for Newton's opinion about a controversy with Christopher Wren and Robert Hooke about the force needed to cause heavenly bodies to move in ellipses in accord with Kepler's laws. Halley was surprise to hear that Newton had already solved this problem. Halley persuaded Newton to write down the solution which Halley arranged to publish as the Principia in 1687. The synthesis of astronomy and physics in this work quickly established Newton as one of the great thinkers of all time.
In this experiment, we shall attempt to predict the path of a comet using Newton's law of universal gravity.
Until Newton's time, observers of comets believed they were earthly phenomena, below the heavens because they moved rapidly past the stars and were not permanent. But Halley and others identified that comets are heavenly objects, often with periodic reappearances.According to Newton, any force on an object not countered by an opposing force will cause a proportional acceleration, a, inversely related to that object's mass:
But it would be difficult for us to continuously calculate the constantly changing acceleration and simultaneously plot the orbit. So we will elect to approximate the path of the comet by applying 60 days worth of acceleration all at once, then letting the comet coast ahead while we calculate the next acceleration.
The only variable controlling the change in the comet's velocity (direction and speed) is distance to the Sun. So to predict the path of a comet, we need to construct a graph of acceleration compared to the distance. Use the following information based on Newton's equations.
real sizeusing either inches or centimeters so that the specified 1.87 inches = 4.75 cm really is that length on your graph paper.
gpredicting the comet's path.