Transcript

1Physics 1100 – Spring 2012

Chapter 10 – Projectile and Satellite MotionChapter 10 – Projectile and Satellite Motion

• Projectile MotionProjectile Motion– Projectiles Launched Horizontally

– Upwardly Launched Projectiles

• Fast Moving Projectiles – SatellitesFast Moving Projectiles – Satellites• Circular Satellite OrbitsCircular Satellite Orbits• Elliptical OrbitsElliptical Orbits• Kepler’s Laws of Planetary MotionKepler’s Laws of Planetary Motion• Energy Conservation and Satellite MotionEnergy Conservation and Satellite Motion• Escape VelocityEscape Velocity

2Physics 1100 – Spring 2012

Gravitational Force is Acting All the Time!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?

3Physics 1100 – Spring 2012

Components of MotionComponents of Motion

• Break the motion into 2 aspects, “components”Break the motion into 2 aspects, “components”– Horizontal– Vertical

• Is there a Is there a force acting in the horizontal direction? acting in the horizontal direction?• Is there a force acting in the vertical direction?Is there a force acting in the vertical direction?• Does the ball accelerate in the horizontal direction?Does the ball accelerate in the horizontal direction?

– Does its horizontal velocity change?• Does the ball accelerate in the vertical direction?Does the ball accelerate in the vertical direction?

– Does its vertical velocity change?

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Analyzing Projectile MotionAnalyzing Projectile Motion

• By breaking the motion into independent parts, analysis is simplified!

• The horizontal and vertical motions are independent

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ProjectilesProjectiles

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Projectile MotionProjectile 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)

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VectorsVectors

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Projectile MotionProjectile Motion

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Class ProblemClass 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)

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Class ProblemClass 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.

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Going into OrbitGoing 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

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Newton’s classic picture of orbitsNewton’s classic picture of orbits

• Low-earth-orbit takes 88 Low-earth-orbit takes 88 minutes to come around full minutes to come around full circlecircle

• Geosynchronous satellites take Geosynchronous satellites take 24 hours24 hours

• The moon takes a monthThe moon takes a month

• Can figure out circular orbit Can figure out circular orbit velocity by setting velocity by setting FFgravitygravity = =

FFcentripetalcentripetal

http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html

13Physics 1100 – Spring 2012

Space Shuttle OrbitSpace 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!

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Geo-synchronous OrbitGeo-synchronous Orbit

• Altitude chosen so that period of orbit = 24 hrsAltitude 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!Stays above the same spot on the Earth!• Only equatorial orbits workOnly equatorial orbits work

– That’s the direction of earth rotation

• Cluttered!Cluttered!– 2,200 in orbit

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KeplerKepler (1600's)(1600's)

• Described the shape of Described the shape of planetary orbitsplanetary orbits as well as their orbital speedsas well as their orbital speeds

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Kepler’s LawsKepler’s Laws

• These are three laws of physics that relate to planetary These are three laws of physics that relate to planetary orbits.orbits.

• These were empirical laws. These were empirical laws.

• Kepler could not explain them. Kepler could not explain them.

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1. Law of Ellipses

The orbits of planets are ellipses with the Sun at one focus

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2. Law of Equal Areas2. Law of Equal Areas

A line joining a planet to the Sun sweeps out equal areas in equal intervals of time

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3. Kepler’s 33. Kepler’s 3rdrd Law Law

The ratio of the square of a planet's orbital period to the cube of its average orbital radius is constant

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Elliptical OrbitsElliptical Orbits

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Newtonian MechanicsNewtonian Mechanics

• Newton introduced the concept of a ‘force’, something Newton introduced the concept of a ‘force’, something that acts to change the motion of matterthat acts to change the motion of matter

• Newton’s gravitational force explained the motions of the planets, and agreed completely with Kepler’s laws

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Class ProblemClass Problem

• The boy on the tower throws a ball 20 meters downrange as The boy on the tower throws a ball 20 meters downrange as shown. What is his pitching speed? shown. What is his pitching speed?

1) 10 m/s 2) 20 m/s 3) 40 m/s 4) 80 m/s 5) 100m/s

23Physics 1100 – Spring 2012

Class ProblemClass Problem

• The boy on the tower throws a ball 20 meters downrange as The boy on the tower throws a ball 20 meters downrange as shown. What is his pitching speed? 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.

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Class ProblemClass 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?

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Class ProblemClass Problem

• Consider two satellites in orbit about Consider two satellites in orbit about a star (like our sun). If one satellite a star (like our sun). If one satellite is twice as far from the star as the is twice as far from the star as the other, but both satellites are other, but both satellites are attracted to the star with the same attracted to the star with the same gravitational force, how do the gravitational force, how do the masses of the satellites compare? masses of the satellites compare?

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Class ProblemClass Problem

• Consider two satellites in orbit about Consider two satellites in orbit about a star (like our sun). If one satellite a star (like our sun). If one satellite is twice as far from the star as the is twice as far from the star as the other, but both satellites are other, but both satellites are attracted to the star with the same attracted to the star with the same gravitational force, how do the gravitational force, how do the masses of the satellites compare? 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.

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