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 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