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Kepler ’ s Laws of Planetary Motion

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(COMPARING PLANETS)

- Kepler’s Third Law:
P: orbital period of planet (sidereal period)

A: planet’s average distance from Sun (semi-major axis)

Sun

A

applet

1 AU (Astronomical Unit) is Earth’s average distance from Sun

1 AU = 1.5 × 1011 m

applet

applet - defunct

If it takes Eris 557 years to orbit the Sun, what is its average distance from the Sun?

(Enter your answer rounded to the nearest whole number of AU.)

- Jupiter:
- Eris:

- Haumea:

The asteroid Apophis has an orbit with the following characteristics:

- semi-major axis: A = 0.922 AU
- orbit period: P = 0.89 yr
- eccentricity: e = 0.191
- inclination: i = 3.3º
If Earth’s orbit has a semi-major axis of 1 AU and Venus has a semi-major axis of 0.723 AU, does Apophis’ orbit cross the planet orbits?

- Apophis always stays between Earth’s and Venus’ orbits.
- Apophis goes outside Earth’s orbit and inside Venus’ orbit.
- Apophis goes outside Earth’s orbit.
- Apophis goes inside Venus’ orbit.

- asteroid about 300 m across
- close approach to Earth in 2029

For Friday, Feb. 21

Reading: Appendix 7

Assignments: Homework #3 (due Wed. Feb 26)

- spacecraft lands and fires rocket to push asteroid
- what direction will the asteroid go?
- what if the asteroid is spinning?
- how far in advance of a collision do we need to start?

VELOCITY

- speed: how fast an object’s position changes
- measured by speedometers, radar guns
- velocity: speed and direction of travel
- measured by weather vanes

- First Law: An object will maintain a constant velocity if there is no net force acting on it.

- acceleration: how fast velocity changes
- 3 ways to accelerate a car:
GAS PEDAL (change speed)

BRAKE PEDAL

STEERING WHEEL (change direction)

- force: strength and direction of a push or pull
- any effort that can cause acceleration

- Second Law: For an unbalanced force,
a: acceleration (units: )

m: mass(units: kg)

F: force (units: Newton = )

MASS REDUCES ACCELERATION

trying to blow up a BIG asteroid is not a great idea…

… even a big explosion wouldn’t accelerate it much Newton’s Second Law

… fragments would keep moving in mostly the same direction Newton’s First Law

A ball is attached to a string and swung in a circular path above my head. At the point shown below, I suddenly release the string. If this is viewed from directly above, which of the paths below would the ball most closely follow when released?

D

E

C

B

A

VIEW FROM ABOVE:

Newton’s First Law says: If object travels on a curved path, there MUST BE an unbalanced force.

FORCE

TOP VIEW:

(friction between tires and road)

VELOCITY

VELOCITY

FORCE

(gravity)

PATH

PATH

Newton’s Second Law says: object accelerates (turns) in direction of unbalanced force

force is NOT pushing planet forward

force IS pulling toward inside of orbit (toward Sun)

Video

A rotating tube makes it feel like there is gravity…

Your body tries to move in a straight line (Newton’s 1st), but floor exerts a force to change your direction…

For Monday, Feb. 24

Reading: Section 4.1, 4.2 (including Math Tools 4.1, 4.2)

Assignments: Homework #3 (due Wed. Feb. 26)

54 km

The picture below shows the velocity of a planet at different times in its orbit (larger arrow means larger speed).

2

Draw the direction of the force on the planet at the different positions shown.

1

For position #3, which combination of these describes the direction of the force is acting:

To the right of the velocity arrow

To the left of the velocity arrow

Forward (with the velocity arrow)

Back (against the velocity arrow)

Enter all that apply.

3

PLANET’S VELOCITY

(ALWAYS TANGENT TO ORBIT)

SUN’S FORCE TURNS AND SPEEDS PLANET

SUN’S FORCE JUST TURNS PLANET

SUN’S FORCE TURNS AND SLOWS PLANET

PLANET’S VELOCITY

THE FORCE POINTS TOWARD THE SUN!

USA

- Third Law: When one object exerts a force on a second object, second object exerts an equal and opposite force back on the first.

EXAMPLES:

GAS FORCE ON ROCKET

ROCKET’S FORCE ON GAS

SKATER FORCES ON EACH OTHER

ICE

Newton’s Third Law: star moves slightly as planet orbits

“wobble” of star depends on star and planet masses, and size of planet orbit

PLANET’S FORCE ON STAR

STAR’S FORCE ON PLANET

A compact car and a large truck collide head-on and stick together.

- Which one feels the largest force during the collision?
- Which one receives the largest acceleration?
(Enter both letters of your answers, then hit “Enter”.)

- The car.
- The truck.
- Both experience the same amount.
- You can’t tell without knowing how fast they were moving before the collision.

FORCE

FORCE

Forces have equal strength, but accelerations may differ:

MAN’S FORCE ON BOAT

BOAT’S FORCE ON MAN

MORE MASS, LESS ACCELERATION

- Moon: 7 1022 kg
1/80th Earth

- Earth: 6 1024 kg
- Jupiter: 2 1027 kg
300 Earth

- Sun: 2 1030 kg
1000 Jupiter

300,000 Earth

.

center of mass

“balance point” is closer to more massive object:

- If center of mass is to remain between two moving objects, less massive object must move faster in exact opposite direction

- From Newton’s 3rd Law:

The Earth has about 80 times the mass of the Moon, and the distance to the Moon is about 60 Earth radii. How far is the center of mass from the center of Earth?

center of mass?

- satellite uses rocket to hover near asteroid
- What direction will the satellite pull the asteroid?
- How long would it take to deflect the asteroid enough?

- Fg: force
- m1, m2: masses
- d: distance between centers of objects
- G: universal gravitational constant
- attractive force: always pulls masses together
- equal strength forces pull on both masses

USA

At which positions does a rocket feel a greater gravitational force from Earth than from the Moon? (There may be more than one answer.)

A

B

C

A hypothetical planet system has planets in equally-spaced circular orbits. The planet masses are given in terms of the mass of the innermost planet. Which of the planets exerts the greatest gravitational force on the star?

A.

B.

C.

D.

16 M

4 AU

10 M

3 AU

3 M

2 AU

1 M

1 AU