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Chapter 10 – Projectile and Satellite Motion • Projectile Motion – Projectiles Launched Horizontally – Upwardly Launched Projectiles • • • • • • Fast Moving Projectiles – Satellites Circular Satellite Orbits Elliptical Orbits Kepler’s Laws of Planetary Motion Energy Conservation and Satellite Motion Escape Velocity Physics 1100 – Spring 2012 1 Gravitational Force is Acting All the Time! • • Consider a tossed ball.... Does gravity ever switch off? As a ball travels in an arc, does the gravitational force change? Physics 1100 – Spring 2012 2 Components of Motion • • • • • Break the motion into 2 aspects, “components” – Horizontal – Vertical Is there a force acting in the horizontal direction? Is there a force acting in the vertical direction? Does the ball accelerate in the horizontal direction? – Does its horizontal velocity change? Does the ball accelerate in the vertical direction? – Does its vertical velocity change? Physics 1100 – Spring 2012 3 Analyzing Projectile Motion • • By breaking the motion into independent parts, analysis is simplified! The horizontal and vertical motions are independent Physics 1100 – Spring 2012 4 Projectiles Physics 1100 – Spring 2012 5 Projectile Motion • • All objects released at the same time (with no vertical initial velocity) will hit the ground at the same time, regardless of their horizontal velocity The horizontal velocity remains constant throughout the motion (since there is no horizontal force) Physics 1100 – Spring 2012 6 Vectors Physics 1100 – Spring 2012 7 Projectile Motion Physics 1100 – Spring 2012 8 Class Problem • When the ball at the end of the string swings to its lowest point, the string is cut by a sharp razor. Which path will the ball then follow? (1) (2) (3) Physics 1100 – Spring 2012 9 Class Problem • When the string is cut, the ball is moving horizontally. After the string is cut there are no forces horizontally, so the ball continues horizontally at constant speed. But there is the force of gravity which causes the ball to accelerate downward, so the ball gains speed in the downward direction. The combination of a constant horizontal speed and a downward gain in speed produces the curved path called a parabola. The ball continues along path b — a parabolic path. Physics 1100 – Spring 2012 10 Going into Orbit • Launch sideways from a mountaintop • If you achieve a speed v such that the force of gravity provides the exact centripetal acceleration need to keep the projectile moving in a circle, the projectile would orbit the Earth at the surface! • How fast is this? – v 8000 m/s = 8 km/s = 28,800 km/hr ~ 18,000 mph Physics 1100 – Spring 2012 11 Newton’s classic picture of orbits • • • • Low-earth-orbit takes 88 minutes to come around full circle Geosynchronous satellites take 24 hours The moon takes a month Can figure out circular orbit velocity by setting Fgravity = Fcentripetal http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html Physics 1100 – Spring 2012 12 Space Shuttle Orbit • Example of LEO, Low Earth Orbit ~200 km altitude above surface • Period of ~90 minutes, v = 7,800 m/s • Decays fairly rapidly due to drag from small residual gases in upper atmosphere – Not a good long-term parking option! Physics 1100 – Spring 2012 13 Geo-synchronous Orbit • Altitude chosen so that period of orbit = 24 hrs – Altitude = 36,000 km (~ 6 R), v = 3,000 m/s • Stays above the same spot on the Earth! • Only equatorial orbits work – That’s the direction of earth rotation • Cluttered! – 2,200 in orbit Physics 1100 – Spring 2012 14 Kepler (1600's) • Described the shape of planetary orbits as well as their orbital speeds Physics 1100 – Spring 2012 15 Kepler’s Laws • These are three laws of physics that relate to planetary orbits. • These were empirical laws. • Kepler could not explain them. Physics 1100 – Spring 2012 16 1. Law of Ellipses The orbits of planets are ellipses with the Sun at one focus Physics 1100 – Spring 2012 17 2. Law of Equal Areas A line joining a planet to the Sun sweeps out equal areas in equal intervals of time Physics 1100 – Spring 2012 18 3. Kepler’s 3rd Law The ratio of the square of a planet's orbital period to the cube of its average orbital radius is constant Physics 1100 – Spring 2012 19 Elliptical Orbits Physics 1100 – Spring 2012 20 Newtonian Mechanics • Newton introduced the concept of a ‘force’, something that acts to change the motion of matter • Newton’s gravitational force explained the motions of the planets, and agreed completely with Kepler’s laws Physics 1100 – Spring 2012 21 Class Problem • The boy on the tower throws a ball 20 meters downrange as shown. What is his pitching speed? 1) 10 m/s 2) 20 m/s 3) 40 m/s 4) 80 m/s Physics 1100 – Spring 2012 5) 100m/s 22 Class Problem • The boy on the tower throws a ball 20 meters downrange as shown. What is his pitching speed? Use the equation for speed as a "guide to thinking.“ v = d/t d is 20m; but we don't know t… the time the ball takes to go 20m. But while the ball moves horizontally 20m, it falls a vertical distance of 4.9m, which takes 1 second… so t = 1s. Physics 1100 – Spring 2012 23 Class Problem • Consider the various positions of the satellite as it orbits the planet as shown. With respect to the planet, in which position does the satellite have the maximum a) speed? b) velocity? c) kinetic energy? d) gravitational potential energy? e) total energy? Physics 1100 – Spring 2012 24 Class Problem • Consider two satellites in orbit about a star (like our sun). If one satellite is twice as far from the star as the other, but both satellites are attracted to the star with the same gravitational force, how do the masses of the satellites compare? Physics 1100 – Spring 2012 25 Class Problem • Consider two satellites in orbit about a star (like our sun). If one satellite is twice as far from the star as the other, but both satellites are attracted to the star with the same gravitational force, how do the masses of the satellites compare? If both satellites had the same mass, then the one twice as far would be attracted to the star with only one-fourth the force (inverse-square law). Since the force is the same for both, the mass of the farthermost satellite must be four times as great as the mass of the closer satellite. Physics 1100 – Spring 2012 26