ONE EQUATION, TWO CONICS, THREE MEN.
______________________________
or, How to use advanced algebra to make sense of the universe!
Johannes Kepler (1571 - 1630) described planetary motion using which conic section? ___________________
Galileo Galilei (1564 - 1642) described projectile motion using which conic section? _____________________
Isaac Newton (1643 – 1727), born within a year of Galileo’s death united the motion of planets and the motion of
projectiles by a single, universal law.
Newton used a thought experiment imagining a cannon at the top of an extremely tall mountain:
Isaac Newton discussed the use of a cannon to place an object in orbit in his Principia Mathematica
(1687) – the book that defined classical physics and provided the theoretical basis for space travel
and rocketry. Newton used the following thought experiment to explain the principle of Earth orbits.
Imagine a mountain so high that its peak is above Earth's atmosphere; on top of this mountain is a
cannon that fires horizontally. As more and more charge is used with each shot, the speed of the
cannonball will be greater, and the projectile will fall to the ground farther and farther from the
mountain. Finally, at a certain speed, the cannonball will not hit the ground at all but will fall toward
the circular Earth just as fast as Earth curves away from it. In the absence of drag from the
atmosphere, it will continue forever in Earth orbit.
(http://www.daviddarling.info/encyclopedia/N/Newton_cannon.html)
http://waowen.screaming.net/revision/force&motion/newtmtn.gif
INVESTIGATE NEWTON’S THOUGHT EXPERIMENT USING THE FOLLOWING APPLET:
http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/newt/newtmtn.html
1. List one initial velocity for which the cannonball will hit the earth between points…
D & E: _________
E & F: _________ F&B: _____________
2. List one initial velocity for which the cannonball orbits once and hits the cannon.
3. List one initial velocity for which the cannonball sails off into space.
Galileo knew the cause of projectile motion was the force of _______________________________.
Kepler knew the shape of planetary orbits were ellipses, but he wasn’t certain of the cause.
Newton now showed that the same force that caused _________________________ motion also caused
planetary motion.
“...from the same principles, I now
demonstrate the frame of the System of
the World.”
-Principia Mathematica.
history.mcs.st-and.ac.uk/PictDisplay/Newton.html
http://www-
The force was quantified by one grand equation, the law of universal gravitation.
This law states that any two bodies in the universe, with masses m1 and m 2 , attract (pull on) each
other with a force that is ________________________________ proportional to the _______________________________
of the distance between their centers. The grand equation?
Fg  G
m1m 2
d2
(mass in kg, distance in m, G is a constant, 6.673  10 11
The units of force, appropriately, are called _______________________________!
N m2
.)
kg 2
ONE NEWTON
OF FORCE IS THE FORCE NEEDED TO GIVE A ONE KILOGRAM MASS AN ACCELERATION OF
ONE METER PER SECOND, PER SECOND.
Use the universal gravitation equation to investigate both PLANETARY ORBITS and PROJECTILE MOTION!
USE A SEPARATE SHEET THAT CLEARLY SHOWS YOUR PROCESS. TRANSFER YOUR FINAL ANSWERS TO THIS SHEET.
YOUR SEPARATE SHEET. Find the sum of the two forces if
this occurs at the earth’s aphelion.
Kepler’s planetary orbits
(refer to planetary data sheet)
1. Find the force keeping Mercury in its orbit
around the sun…
a. when Mercury is at its closest point to the sun
(perihelion):
b. when Mercury is at its farthest point from the sun
(aphelion):
2. Find the force keeping Pluto in its orbit around
the sun…
a. when Pluto is at its perihelion:
b. when Pluto is at its aphelion:
3. Find the force keeping Earth in its orbit around
the sun…
a. when Earth is at its perihelion:
Galileo’s projectile Motion
1. Find the force “pulling down” a 10-kg cannonball at
the moment that it is launched from the surface of the
earth. (note: how far is the cannonball from the center of the
earth?)
2. Find the force “pulling down” the 10-kg cannonball at
the moment that it reaches it’s peak height of 50 meters
in the air. (note: how far is the cannonball from the center of
the earth?) What is the percent change in force from
when it was launched?.
3. Suppose you have the ability to launch the cannonball
to the moon! Find the force drawing the 10-kg
cannonball toward the earth when it is 10,000 km above
the earth’s surface.
b. when Earth is at its aphelion:
4. Find the force keeping the moon in its orbit
around the earth.
5. Sometimes the moon and the sun are
diametrically opposite each other, with the earth in
between. Thus, the sun and moon are pulling in
opposite directions. DIAGRAM THIS ON YOUR SEPARATE
Find the two opposing forces, and the difference
between these forces if this occurs at the earth’s
perihelion. By what percent does the moon’s “pull”
decrease the sun’s “pull”?
4. Find the force drawing the 10-kg cannonball toward
the earth when it is halfway to the moon.
5. At the halfway point, what is the force (exerted by the
moon) drawing the cannonball toward the moon?
SHEET.
FORCES:
DIFFERENCE:
6. On an attached sheet, plot (on one axis system) the
graphs of the force of the earth (Fe) and the force of the
moon (Fm) as functions of the distance (d) from the
surface of the earth. Let “d” start at the surface of the
earth and end when you are on the surface of the moon.
PERCENT:
6. Sometimes the moon and the sun are lined up on
the same side of the earth. Thus, the sun and moon
are pulling in the same direction. DIAGRAM THIS ON
7. At what distance (d) above the earth’s surface is the
pull of the earth equal to (balanced by) the pull of the
moon? Circle and label this point on your plot. Explain
what this means for the cannonball.
AXIS SYSTEM FOR # 6, ABOVE.
Notes:
1. The surface of the earth is 6378 kilometers from the center of the earth.
2. At the surface of the earth, the distance to the surface of the moon is 384,000 km, which would make it 385,738 km to the center of the moon.
Planet
Eccentricity
Perihelion
Distance
closest point to the Sun
(AU)
Aphelion
Distance
farthest point from the
Sun
(AU)
Period of
Revolution
Around the Sun
(1 planetary year)
Period of Rotation
(1 planetary day)
Mass
(kg)
Diameter
(miles
km)
3,031 miles
4,878 km
7,521
miles
Venus
224.68 Earth days
243 Earth days
4.87 x 1024
0.007
0.718
0.728
12,104 km
7,926 miles
Earth
365.26 days
24 hours
5.98 x 1024
0.017
0.98
1.02
12,756 km
24.6 Earth hours
4,222
miles
Mars
686.98 Earth days
6.42 x 1023
0.093
1.38
1.67
=1.026 Earth days
6,787 km
88,729 miles
Jupiter
11.862 Earth years
9.84 Earth hours
1.90 x 1027
0.048
4.95
5.45
142,796 km
74,600
miles
Saturn
29.456 Earth years
10.2 Earth hours
5.69 x 1026
0.056
9.02
10.0
120,660 km
32,600 miles
Uranus
84.07 Earth years
17.9 Earth hours
8.68 x 1025
0.047
18.3
20.1
51,118 km
30,200
miles
Neptune
164.81 Earth years
19.1 Earth hours
1.02 x 1026
0.009
30.0
30.3
48,600 km
1,413 miles
Pluto
247.7 years
6.39 Earth days
1.29 x 1022
0.248
29.7
49.9
2,274 km
Note: The eccentricity of a planet's orbit measures how much it departs from a perfect circle. Orbits with zero eccentricity (e = 0) are circular; orbits with eccentricities
close to 1 (e ~ 1) are long and skinny. Planetary orbits tend to be almost circular while comets and many asteroids follow more eccentric paths.
Mercury
0.206
0.31
0.47
87.96 Earth days
58.7 Earth days
3.3 x 1023
The above data table is taken in part from http://science.nasa.gov/headlines/y2001/ast04jan_1.htm
Notes:
1 AU = 93 million miles = 149.6 million km
the SUN
the MOON
mass: 1.989 X 10^30 kg
mass: 7.35 X 10^22 kg
diameter: 1, 390,000 km
diameter: 3476 km
distance from the earth: 384,000 km