ANNOUNCEMENTS:
• Test repair- Tomorrow 7:20 AM and beginning of 5A
lunch!
• Inuit Yoyo Lab due today
• My Solar System Phet Lab due Friday, 9/11
• Unit 3 ‘Exam” and Unit 3 HW due Friday, 9/11
• Rocket Lab III Monday, 9/14- bring more 2L bottles!
• Field Trips!!!
– Permission Slips
– The Martian: 10/2, $5.50
– UTED: 10/22, $5.00
• Research Day 9/24 7th, 9/25 6th-8th
Unit I-3
Circular Motion and
Gravitation
Part 3: Satellites
& Kepler’s Laws
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Trajectory
• The curved path a projectile follows
• Parabolic
• Two general forms:
Launched Horizontally:
Launched at an Angle:
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Let’s look at the horizontal trajectory some
more:
• What happens to the trajectory as vi is increased?
• The x distance traveled in the amount of time
required to fall the y distance increases.
• Until….?
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Newton’s thought experiment:
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
You get a satellite
• Satellite = a projectile that falls around Earth instead of into it.
• Must have velocity such that it’s range is greater than the
curvature of earth.
• Curvature of Earth is 5 m in 8 km.
• This means the velocity must be at least 8 km/s which is 29 000
km/hr!
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Circular Orbits
• Newton proposed a circular orbit.
• In circular orbit an object moves at constant speed
because gravity acts downwards so there is no force in
the same direction as the motion (No F means no a of
the m!)
• Satellites in higher orbits have slower tangential speeds
and longer periods (time to complete an orbit).
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
If the satellite exceeds the
8 km/s speed you get:
• An elliptical orbit.
• An ellipse is a closed path taken by a point
that moves in such a way that the sum of its
distances from two fixed points, foci, is
constant.
• http://www.mathopenref.com/constellipse1.
html
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Tycho Brahe
• Danish, 1546-1601
• Given Isle of Hven
• Recorded positions of
planets for over 20
years!
• Had a prosthetic
nose- dueling!
• Died of burst bladder!
• Apprenticed Kepler
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Johannes Kepler
• German, 1571-1630
• Applied math to
Brahe’s observations.
• Developed 3 Laws of
Planetary Motion.
• Showed paths of
planets are not
circles. Instead are
ellipses.
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Kepler’s 1st Law
• The path of each planet around the Sun is
an ellipse with the sun at one focus.
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Kepler’s 2nd Law
• A line from the Sun to the planet sweeps out
equal areas of space in equal time intervals.
Kepler’s 3rd Law
• The square of the period of a planet is directly
proportional to the cube of it’s average distance
from the Sun.
• That’s easier to say in math:
3
2
T ar
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Kepler’s 3rd Law
3
2
T ar
• We still don’t have a button for a!
• The constant of proportionality comes from
Universal Gravitation and is equal to 4p2/Gm,
where m is the mass of the central object.
Let’s don’t solve for
period squared!
2
T=
2
4p
Gm
3
r
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
CARD!
Kepler’s 3rd Law
3
r
T= 2p
1/2
Gm
And another convenient form………………
CARD!
m
v= Gr
1/2
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
Kepler’s 3rd Law for 2 objects
This relationship can be extended to two objects circling
the same object (i.e.- 2 planets around the sun)
The ratio of the squares of the periods of the satellites is
equal to the ratio of the cubes of their radii.
Again, easier in math!
2
TA
=
2
TB
2
rA
TA
=
rB
TB
rA
3
rB
3
CARD!
Objective: Use Universal Gravitation and Kepler’s Laws to predict satellite orbits.
3