Phy 101: Fundamentals of Physics I Chapter 10: Projectiles & Satellite Motion Lecture Notes Johannes Kepler (15711630) German mathematician Mathematical contributions: led to the development of calculus Derived the first proof of how logarithms work Discovered 2 regular polyhedrons Astronomy contributions:
Calculated the most exact astronomical tables of his time (using stolen data) Demonstrated that the planets move in elliptical orbits around the sun Coined the term satellite to describe the moons of Jupiter observed by Galileo Projectile Motion 2-dimension motion that occurs at the surface of the earth Key ideas Vertical motion is treated like free fall Horizontal motion is constant Horizontal & vertical components of motion occur
independent of each other Consider a ball that rolls off a flat table-top: Upwardly Launched Projectiles Satellite Motion Toss a ball horizontally: it falls to the ground
Toss the ball harder: It falls further away from you Toss the ball even harder: It falls still further away from you Now, imagine tossing a ball so hard that:
As it falls it follows the curvature of the earth This is referred to as satellite motion As the objects falls: it never reaches the earth It remains in constant free fall v Elliptical Orbits When an object in projectile motion is moving so fast that its fall does not follow the curvature of the earth it will
Overshoot a circular orbit Follow an elliptical orbit The speed of the object will vary while in elliptical orbit: closer to the earth it will move faster Farther from the earth it will move slower Keplers Laws of Planetary Motion Kepler derived a set of generalized statements describing the motion of the planets, based on the observations of Tycho
Brahe These statements are know known as Keplers Laws: 1) Each planet moves in an elliptical orbit with the sun at one focus of the ellipse 2) The line from the sun to any planet sweeps equal areas of space in equal time intervals 3) The squares of times of revolutions (periods) of the planets are proportional to the cubes of their average distances from the sun: T2 ~ R3 {for all planets} Energy Conservation & Satellite Motion When objects maintain satellite motion their energy is conserved: PE + KE = constant value This is true whether the orbit is circular or elliptical
Circular orbit: v and KE will remain constant Elliptical orbit: v and KE will vary When the satellite is closest to the planet (PE is lowest) its speed (& KE) is its fastest value for the orbit When the satellite is farthest from the planet (PE is highest) its speed (& KE) is at its lowest value for the orbit In both cases, KE + PE remains the same value Escape Speed The speed a hurled object must exceed to escape the
gravitational influence of a planet To determine escape speed: vescape = (2.gplanet.d)1/2 Or (refer to Ch 9 for details on gravitational force) vescape = (2.G.M/d)1/2 Question: How fast must an object be moving to escape the earths gravity? Bottomline: Object must have more KE than PE to escape the gravitational influence (field) of a planet!
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