Transcript Kepler`s Laws

Chapter 10 – Projectile and Satellite Motion
• Projectile Motion
– Projectiles Launched Horizontally
– Upwardly Launched Projectiles
•
•
•
•
•
•
Fast Moving Projectiles – Satellites
Circular Satellite Orbits
Elliptical Orbits
Kepler’s Laws of Planetary Motion
Energy Conservation and Satellite Motion
Escape Velocity
Physics 1100 – Spring 2012
1
Gravitational Force is Acting All the Time!
•
•
Consider a tossed ball.... Does gravity ever switch off?
As a ball travels in an arc, does the gravitational force change?
Physics 1100 – Spring 2012
2
Components of Motion
•
•
•
•
•
Break the motion into 2 aspects, “components”
– Horizontal
– Vertical
Is there a force acting in the horizontal direction?
Is there a force acting in the vertical direction?
Does the ball accelerate in the horizontal direction?
– Does its horizontal velocity change?
Does the ball accelerate in the vertical direction?
– Does its vertical velocity change?
Physics 1100 – Spring 2012
3
Analyzing Projectile Motion
•
•
By breaking the motion into independent parts, analysis is simplified!
The horizontal and vertical motions are independent
Physics 1100 – Spring 2012
4
Projectiles
Physics 1100 – Spring 2012
5
Projectile Motion
•
•
All objects released at the same time (with no vertical initial velocity) will hit
the ground at the same time, regardless of their horizontal velocity
The horizontal velocity remains constant throughout the motion (since there
is no horizontal force)
Physics 1100 – Spring 2012
6
Vectors
Physics 1100 – Spring 2012
7
Projectile Motion
Physics 1100 – Spring 2012
8
Class Problem
•
When the ball at the end of the string swings to its lowest point, the
string is cut by a sharp razor. Which path will the ball then follow?
(1)
(2)
(3)
Physics 1100 – Spring 2012
9
Class Problem
•
When the string is cut, the ball is moving horizontally. After the string is
cut there are no forces horizontally, so the ball continues horizontally at
constant speed. But there is the force of gravity which causes the ball
to accelerate downward, so the ball gains speed in the downward
direction. The combination of a constant horizontal speed and a
downward gain in speed produces the curved path called a parabola.
The ball continues along path b — a parabolic path.
Physics 1100 – Spring 2012
10
Going into Orbit
• Launch sideways from a mountaintop
• If you achieve a speed v such that the force of gravity provides
the exact centripetal acceleration need to keep the projectile
moving in a circle, the projectile would orbit the Earth at the
surface!
• How fast is this?
– v  8000 m/s = 8 km/s = 28,800 km/hr ~ 18,000 mph
Physics 1100 – Spring 2012
11
Newton’s classic picture of orbits
•
•
•
•
Low-earth-orbit takes 88
minutes to come around full
circle
Geosynchronous satellites take
24 hours
The moon takes a month
Can figure out circular orbit
velocity by setting Fgravity =
Fcentripetal
http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html
Physics 1100 – Spring 2012
12
Space Shuttle Orbit
• Example of LEO, Low Earth Orbit ~200 km altitude above
surface
• Period of ~90 minutes, v = 7,800 m/s
• Decays fairly rapidly due to drag from small residual gases in
upper atmosphere
– Not a good long-term parking option!
Physics 1100 – Spring 2012
13
Geo-synchronous Orbit
• Altitude chosen so that period of orbit = 24 hrs
– Altitude = 36,000 km (~ 6 R), v = 3,000 m/s
• Stays above the same spot on the Earth!
• Only equatorial orbits work
– That’s the direction of earth rotation
• Cluttered!
– 2,200 in orbit
Physics 1100 – Spring 2012
14
Kepler
(1600's)
•
Described the shape of
planetary orbits
as well as their orbital speeds
Physics 1100 – Spring 2012
15
Kepler’s Laws
• These are three laws of physics that relate to planetary
orbits.
• These were empirical laws.
• Kepler could not explain them.
Physics 1100 – Spring 2012
16
1. Law of Ellipses
The orbits of planets are ellipses with the Sun at one focus
Physics 1100 – Spring 2012
17
2. Law of Equal Areas
A line joining a planet to the Sun sweeps out equal areas in equal
intervals of time
Physics 1100 – Spring 2012
18
3. Kepler’s 3rd Law
The ratio of the square of a planet's orbital period to
the cube of its average orbital radius is constant
Physics 1100 – Spring 2012
19
Elliptical Orbits
Physics 1100 – Spring 2012
20
Newtonian Mechanics
• Newton introduced the concept of a ‘force’, something that acts
to change the motion of matter
• Newton’s gravitational force explained the
motions of the planets, and agreed
completely with Kepler’s laws
Physics 1100 – Spring 2012
21
Class Problem
•
The boy on the tower throws a ball 20 meters downrange as
shown. What is his pitching speed?
1) 10 m/s
2) 20 m/s 3) 40 m/s 4) 80 m/s
Physics 1100 – Spring 2012
5) 100m/s
22
Class Problem
•
The boy on the tower throws a ball 20 meters downrange as
shown. What is his pitching speed?
Use the equation for speed as a "guide to thinking.“
v = d/t
d is 20m; but we don't know t… the time the ball takes to go 20m. But while the
ball moves horizontally 20m, it falls a vertical distance of 4.9m, which takes 1
second… so t = 1s.
Physics 1100 – Spring 2012
23
Class Problem
•
Consider the various positions of
the satellite as it orbits the planet as
shown. With respect to the planet, in
which position does the satellite
have the maximum
a) speed?
b) velocity?
c) kinetic energy?
d) gravitational potential energy?
e) total energy?
Physics 1100 – Spring 2012
24
Class Problem
•
Consider two satellites in orbit about
a star (like our sun). If one satellite
is twice as far from the star as the
other, but both satellites are
attracted to the star with the same
gravitational force, how do the
masses of the satellites compare?
Physics 1100 – Spring 2012
25
Class Problem
•
Consider two satellites in orbit about
a star (like our sun). If one satellite
is twice as far from the star as the
other, but both satellites are
attracted to the star with the same
gravitational force, how do the
masses of the satellites compare?
If both satellites had the same mass, then the one twice as far would be
attracted to the star with only one-fourth the force (inverse-square law).
Since the force is the same for both, the mass of the farthermost satellite
must be four times as great as the mass of the closer satellite.
Physics 1100 – Spring 2012
26