projectile and satellite motion projectile motion we choose to break up projectile motion as a...
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Projectile and Satellite Motion
PROJECTILE MOTION
•We choose to break up Projectile Motion as a combination of vertical free-fall motion and horizontal motion at a constant speed. Forces are at work in only the vertical
motion
Parabola (approximately)
For a particular range less than the maximumand for a particular launch velocity,two different launch angles will give that range.The two angles add to give 900.450 gives the maximum range.
Simulation
Free Fall Results
•Horizontal motion is constant No force is acting
•Vertical Motion is accelerated Gravity acts in this direction
•Rise time = Fall Time
•Final speed = Initial speed
•Actual path and ideal path (free fall) are different.
FAST-MOVING PROJECTILES - SATELLITES
• Let’s throw stones horizontally with
ever increasing velocity.
• The Earth’s curvature is 16 ft for every
5 miles (4.9 m for 8 km).
16 ft
16 ftCurvature of Earth
5 miles
Throw an object faster, faster.faster,
We draw in each trajectory for 1 second
Kepler’s Laws
•First LawFirst LawPlanets orbit the Sun in ellipses with the Sun at one focus of the ellipse
Sun
Planet
Ellipses
Focus Focus
d1d2
d1 + d2 = constant for any point on ellipse
Ellipses
a = Semi-major axis
b = Semi-minor axis
Eccentricity
a
c
e = c/a
Kepler’s Laws
•Second LawA line drawn from the planet to
the Sun sweeps out equal areas in equal intervals of time
The Search for Order
•Perfect solids
The Search for Order
•Music of the Spheres
Kepler’s Laws
•Third LawThe orbital period of a planet squared is proportional to the length of the semi-major axis cubed.
P2 a3
Using the Third Law
P2 a3
P2 constanta3
P2 a3
P measured in years, a in AU, object orbits Sun
Kepler’s Laws
•Empirical
•Kepler could not explain why the planets orbited the Sun (he thought it had something to do with magnetism)
•Universal
The Apple
2
R
mGMF
m
M
mgR
mGM
2
2
R
GMg
The Earth and Moon
F
2R
mGMF
Earth
Moon
R
maR
mGM
2
G
aRM
2
The free-fall motion of a projectile near the surface of the earth is very nearly parabolic.A parabola is one of the conic sections.
Circle Ellipse Parabola Hyperbola
Ellipse
Ellipse
Ellipse - Circle
Ellipse
Ellipse
Parabola
Vcircle = 5 mi/s = 8 km/s Vescape = 7 mi/s = 11 km/s
Hyperbola
Orbiting
•Falling without getting closer to the ground.
•Force of Gravity = Centripetal Force
d
GMv
vd
GMd
mv
d
GMm
orb
2
2
2