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Chapter 6: Gravity & Projectile Motion
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• Chapter 6: Gravity & Projectile Motion

• Sun at Center Orbits are Circular

• Tycho Brahe1546-1601

Tycho was the greatest observational astronomer of his time. Tycho did not believe in the Copernican model because he didn’t believe that the Earth Moved.

• Kepler worked for Tycho as his mathematician. Kepler derived his laws of planetary motion from Tycho’s observational data.

Orbits are Elliptical

• Kepler’s 3 Laws of Planetary Motion

1: The orbit of each planet about the sun is an ellipse with the sun at one focus.

2. Each planet moves so that it sweeps out equal areas in equal times.

2 21 12 2

2 2

3. T rT r

=

• Isaac Newton(1642 -1727)

• Universal Law of GravityUniversal Law of GravityEvery particle in the Universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product to their masses and inversely proportional to the square of the distance between them.

2~mMFd

d M

m

• Gravitational Force is Universal

The same force that makes the apple fall to Earth, causes the moon to fall around the Earth.

• Universal Law of GravityUniversal Law of Gravity

d M

m

211

26.67 10NmG xkg

−=

2

GmMFd

=

• Measuring G: Cavendish

211

26.67 10NmG xkg

−=

2~mMFd 2

GmMFd

=

G is the same everywhere in the Universe.

• Inverse Square Law

2

GmMFd

=

• Gravitational ForceHow would the force of gravity and the acceleration due to

gravity change as you fell through a hole in the Earth?

• ProblemCalculate the acceleration of gravity acting on you at the

surface of the Earth.

2you EGm MFd

= youF m a=

2EGMa

d=

( )( )( )

11 2 2 24

28

6.673 10 / 5.98 10

3.84 10

x Nm kg x kga

x m

=

29.81 /a m s=

• ProblemCalculate the force of gravity between the Earth and the Moon.

Assume a circular orbit.

2

GmMFd

=

( )( )( )( )

11 2 2 24 22

28

6.673 10 / 6.00 10 7.4 10

3.84 10

x Nm kg x kg x kgF

x m

=

202.01 10F x N=

• ProblemCalculate the acceleration of the Earth due to the Earth-Moon

gravitational interaction.

Fam

=

20

24

2.01 105.98 10

x Nax kg

=

5 23.33 10 /a x m s−=

• Problem

Fam

=

20

22

2.01 107.35 10

x Nax kg

=

3 22.73 10 /a x m s−=

Calculate the acceleration of the Moon due to the Earth-Moon gravitational interaction.

• ProblemThe acceleration of gravity at the Moon due to the Earth is:

3 22.73 10 /a x m s−=

5 23.33 10 /a x m s−=

The acceleration of gravity at the Earth due to the moon is:

Why the difference?

FORCE is the same. Acceleration is NOT!!!

• Force is not Acceleration!

The forces are equal but the accelerations are not!

Earth on Moon Moon on EarthF F= −

• Gravitational forces produce the tides and keep the same side of the Moon always facing

the Earth in a Synchronous Rotation.

• The gravitational force between Earth and Moon results in tides

Large spring tides occur when the Sun and Moon are aligned such that they BOTH pull on the Earth’s water.

Smaller neap tides occur when the Sun and Moon are misaligned such that they pull on the Earth’s water in different directions.

• Earth’s Ocean Tides

Spring Tides

Neap Tides

• How many High and Low Tides per day?

• Earth- Moon System: Tidal LockMoon pulls on Earth and causes tidal bulges in Oceans. Earth pulls on moon causing tidal bulge in Moon such that the CM is off from CG.

Earth’s pull on the moon at its center of gravity produces a torque which rotates it. The moon is ‘tidally locked’ so that it rotates at the same rate at which it revolves, showing Earth only one face.

• Questions!

The Sun exerts a greater force on the Earth. It also causes tides, though it has about half the effect as the moon. Tides are due to differences in force on opposite sides of the Earth. The sun is so far away relative to the size of the Earth that the difference in force is not as significant as the difference due to the Moon since it is closer!

Which exerts a greater gravitational force on the Earth, the moon or the Sun? Which has a greater effect on the Earth’s tides? Why?

• Spring

Spring

Neap

• Bouncing laser beams off the

Moon demonstrates that it slowly moving away from the Earth~.25 cm/month

• Earth- Moon System is Changing!

•Earth Rotation Slowing due to friction of ocean on bottom•.0023 s per century: 900 Million yrs ago, Earth day was 18 hrs! •Decrease of Earth’s spin angular momentum, increases the orbital angular momentum of the Moon!•Earth is slowing down and Moon is moving further away!

• Eventually the Earth's rotation period will be identical to the Moon's orbital period. This situation is called synchronous (1:1) rotation. In the distant future (many billions of years from now), the Earth will have a day which is 47 current days long, and the Moon will only be visible from one side of the Earth!!

• Projectile MotionIgnore Air Resistance!

Most Important:X and Y components are INDEPENDENT of each other!

• Zero at the Top!Y component of velocity is zero at the top of the path in both cases!

You know they have the y component of velocity is the same in both cases because they reached the same height!

• Horizontal and vertical components are independent of each other!

Gravity acts in the vertical direction but not in the horizontal direction!!

Speed in vertical direction speeds up!Speed in horizontal direction stays the same!

Projectile Motion

• First the SIMPLE Case: Horizontal Launch

The x-component doesn’t change (no acceleration in x-direction.)The y-component changes (a = -g.)

(Ignore Air Resistance)

• No Change

change

0xa =

ya g=

Actual path is a vector sum of horizontal and vertical motions.

Projectile Motion

• Plane and PackageAn airplane traveling at a constant speed and height drops a care package. Ignoring air resistance, at the moment the package hits the ground, where is it relative to the plane?

a) Behind the plane.b) Under the plane.c) In front of the plane.

Dropping From Moving Frame

Any object dropped from a plane has the same initial velocity as the plane!

• Projectile MotionWhich hits the ground first? SAME!

If it takes 1 second to hit the ground, how tall is the tower?

Both falling the same height! Horizontal speed doesn’t effect the time to hit the ground!

• Projectile Motion

If it takes 1 second to hit the ground, how tall is the tower?

2 25 5(1) 5d t m= = =

• Same rock, same speed, same angle.Which rock hits the water first?

a) Rock 1 b) Rock 2 c) same

Which rock hits the water with the greatest speed?a) Rock 1 b) Rock 2 c) same

• Which rock hits the water first?a) Rock 1 b) Rock 2 c) same

Which rock hits the water with the greatest speed?a) Rock 1 b) Rock 2 c) same

SpatialSymmetryIn G Field!

Same rock, same speed, same angle.

• Projectile Motion Launched at an Angle

The x-component doesn’t change (no acceleration in x-direction.)The y-component changes (a = -g.)

(Ignore Air Resistance)

• Shooting UP From Moving Frame

Horizontal and vertical components are independent of each other!Horizontal component remains unchanged without air resistance.

Only the vertical component changes!

• Projectiles Launched at an Angle:The simple case: Δy=0

• Question

d

The ball is thrown horizontally at 20 m/s.About how long does it take to hit the ground?

How far does it travel in the horizontal direction?

20 1 20Δ = = =ximx v t s ms

210 /g m s=

212yi

y v t gtΔ = + 2 1yt sgΔ

= =0

• Question

d

The ball is thrown horizontally at 30 m/s.About how long does it take to hit the ground?

How far does it travel in the horizontal direction?

30 1 30Δ = = =ximx v t s ms

210 /g m s=

212yi

y v t gtΔ = + 2 1yt sgΔ

= =

Only Δy determines time!

• QuestionThe ball is thrown horizontally at 100 m/s.

How long does it take to hit the ground? 1 Second!!

How far does it travel in the horizontal direction?

100 1 100Δ = = =ximx v t s ms

210 /g m s=

• Curvature of EarthIf you threw the ball at 8000 m/s off the surface of the Earth

(and there were no buildings or mountains in the way)how far would it travel in the vertical and horizontal

directions in 1 second?

• Curvature of EarthIf you threw the ball at 8000 m/s off the surface of the Earth

(and there were no buildings or mountains in the way)how far would it travel in the vertical and horizontal

directions in 1 second?

( )( ): 8000 / 1 8000Δ = = =xhorizontal x v t m s s m

( )22 21: 5 5 1 52

Δ = = = =vertical y gt t s m

• Curvature of EarthIf you threw the ball at 8000 m/s off the surface of the Earth

(and there were no buildings or mountains in the way)how far would it travel in the vertical and horizontal

directions in 1 second?

( )( ): 8000 / 1 8000Δ = = =xhorizontal x v t m s s m

( )22 21: 5 5 1 52

Δ = = = =vertical y gt t s m

• Curvature of EarthIf you threw the ball at 8000 m/s off the surface of the Earth

(and there were no buildings or mountains in the way)how far would it travel in the vertical and horizontal

directions in 1 second?

Does the ball ever hit the Earth????

• Curvature of EarthCurvature of the Earth: Every 8000 m,

the Earth curves by 5 meters!

• Orbital VelocityIf you can throw a ball at 8000m/s, the Earth curves away

from it so that the ball continually falls in free fall around the Earth – it is in orbit around the Earth!

Above the atmosphere

Ignoring air resistance.

• Orbital Motion| & Escape Velocity8km/s: Circular orbit

Between 8 & 11.2 km/s: Elliptical orbit11.2 km/s: Escape Earth

42.5 km/s: Escape Solar System!

• Projectile Motion IS Orbital MotionThe Earth is in the way!

• Circular Orbital VelocityThe force of gravity is perpendicular to the velocity of the ball so it doesn’t speed it up – it changed the direction of the ball.It provide a centripetal acceleration the keeps it in a circle!

At a speed of 8km/sOrbital time is ~ 90 minutes

• Elliptical OrbitsDoes the force of gravity change the speed of on object in an elliptical orbit?

Because there is a component of force in the direction of motion!

Do Workbook 45-47

• Orbits

Circular Orbit Elliptical Orbit

Where does the satellite move faster in the elliptical orbit?Near the Earth or far from Earth? Why?

• Geosynchronous Orbit

• Elliptical Orbit

• Satellite Orbits

• Global Geostationary Satellite Coverage

USSR China

Japan

Euro

USAUSA

• With an orbital period of about 100 minutes, these satellites will complete slightly more than 14 orbits in a single day.

• Sun-Synchronous Near Polar Orbits

• Orbiting Space Trash

More than 4,000 satellites have been launched into space since 1957. All that activity has led to large amounts of space trash. More than 13,000 objects that are at least three to four inches (seven to ten centimeters) wide. Of those objects, only 600 to 700 are still in use. 95 percent of everything up there that the United States is tracking is trash. There are millions of smaller parts that are too small to track.

Man-made debris orbits at a speed of roughly 17,500 miles/hour (28,000 km/h)!

• Orbiting Space TrashFast Trash Go Boom

Australia, in 1979.

• Orbiting Space TrashWhat Goes Up Must Come Down

This is the main propellant tank of the second stage of a Delta 2 launch vehicle which landed near Georgetown, TX, on 22 January 1997. This approximately 250 kg tank is primarily a stainless steel structure and survived reentry relatively intact.

Skylab crashed ontoAustralia in 1979.

http://www.secretsituation.com/spiltext/factory.htm

• Nuclear Power in Space

• Mission to Mars

The NASA Mars Exploration Program (MEP) will launch a spaceflight mission to Mars in late 2009 that will land a nuclear powered roving Mars Science Laboratory on the surface of the planet.

• “...we’re all astronauts aboard a little spaceship called Earth”- Bucky Fuller

One island in one ocean...from space

Our Spaceship Earth

Chapter 6: Gravity & Projectile MotionTycho Brahe�1546-1601Orbits are EllipticalKepler’s 3 Laws of �Planetary MotionGravitational Force is �UniversalMeasuring G: CavendishInverse Square LawGravitational ForceProblemProblemProblemProblemProblemForce is not Acceleration!Gravitational forces produce the tides and keep the same side of the Moon always facing the Earth in a Synchronous Rotation.The gravitational force between Earth and Moon results in tidesEarth’s Ocean TidesHow many High and Low Tides per day?Earth- Moon System: Tidal LockQuestions!Bouncing laser beams off the Moon demonstrates that it slowly moving away from the Earth�~.25 cm/monthProjectile Motion�Ignore Air Resistance!Zero at the Top!�Y component of velocity is zero at the top of the path in both cases!Projectile MotionFirst the SIMPLE Case: �Horizontal LaunchProjectile MotionPlane and PackageProjectile MotionProjectile MotionSame rock, same speed, same angle.Same rock, same speed, same angle.Projectile Motion �Launched at an AngleShooting UP From Moving Frame�Projectiles Launched at an Angle:�The simple case: y=0QuestionQuestionQuestionCurvature of EarthCurvature of EarthCurvature of EarthCurvature of EarthCurvature of EarthOrbital VelocityOrbital Motion| & Escape VelocityProjectile Motion IS Orbital MotionCircular Orbital VelocityElliptical OrbitsOrbitsGeosynchronous OrbitElliptical OrbitSatellite OrbitsGlobal Geostationary Satellite Coverage �Sun-Synchronous Near Polar OrbitsOrbiting Space Trash�Orbiting Space Trash�Fast Trash Go BoomOrbiting Space Trash�What Goes Up Must Come DownRadioactive Space Junk�Cosmos 954Nuclear Power in SpaceMission to Mars