Transcript Projectile and Satellite Motion

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)
Monkey in a Tree
Set to 48 minutes
For a particular range less than the maximum
and 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).
Throw an object faster, faster, faster.
5 miles
16 ft
We draw in each trajectory for 1 second
Kepler’s Laws
• First Law
Planets
orbit the Sun in ellipses
with the Sun at one focus of the
ellipse
Planet
Sun
Ellipses
d1
Focus
d2
Focus
d1 + d2 = constant for any point on
ellipse
Ellipses
b
a
a = Semi-major axis
b = Semi-minor axis
Eccentricity
c
a
e=
c/a
Kepler’s Laws
• Second Law
A
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 Law
The
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 =(constant)a3
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
m
M
GM  m
F=
2
R
GM  m
= mg
2
R
GM 
g=
2
R
The Earth and Moon
Moon
F
R
GM  m
F=
2
R
GM  m
= ma
2
R
Earth
aR
M =
G
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
Vescape = 7 mi/s = 11 km/s
Vcircle = 5 mi/s = 8 km/s
Ellipse
Ellipse
Ellipse - Circle
Ellipse
Ellipse
Orbiting
• Falling without getting
closer to the ground.
• Force of Gravity =
Centripetal Force
GMm mv
=
2
d
d
GM
2
=v
d
GM
vorb =
d
2