Download 6.1-6.3 Planetary Motion - York Catholic District School Board

6.1-6.3 P LANETARY M OTION
F ORCE
FIELDS

Not the kind you find in Star Trek (the
coolest show ever)

Force fields are used to describe the
amount of a given type of force
generated by an object on other
objects that are near it per a unit
measurement
F OR
EXAMPLE …

A gravitational field affects objects
that have mass

Therefore, any object that possesses
mass that is within a gravitational field
will experience a gravitational force
acting on it

How much force is acting on it will
depend on where this object is – and
how much mass it has
W HY
USE FORCE FIELDS ?
What happens if objects
of the same mass move
further or closer to the
object creating the field?
Is the amount of gravity acting
on two objects of different mass
the same, if their distances from
the earth are the same?
F ORCE FIELDS ARE LIKE
PRICES

What force fields allow you to do is to
calculate the value of a force acting on
an object depending on a set of
conditions

It’s like pricing objects in a store

Is it easier to give the price of one, two
or three objects…

Or give a price per object?
P RICE

A price gives you the cost per item – so
you can predict the cost of a purchase
based on how many items you purchase

Mathematically, a simple way of viewing
a field is looking at the measurement as a
price

The strength of a field at one point tells
you how much the total value will be
based on the amount of a particular
quality that the field affects
TAKE ANOTHER LOOK AT AN
EQUATION YOU KNOW ….
How does:
F = Gm1m2
d2
Define a gravitational field?
See notebook file for derivation of gravitational field
P LANETARY
ORBITS

The orbits of planets are not circular;
they are actually ellipses:

http://mistupid.com/astronomy/orbits
.htm

However, in order to derive the
velocity of orbits based on the
gravitational pull between two bodies
can be dealt with by assuming that the
orbit is circular
C IRCULAR MOTION AND
ORBITS

Recall that when looking at circular
motion, an object maintains a
constant velocity in a circular path if
there is a constant force that pulls the
circulating object towards the center
of its rotation

This situation is similar to how orbits
are formed
C IRCULAR ORBIT AND
PLANETARY MOTION

Compare the relationship between an
orbiting planet and the motion of an object
on a string

http://www.physclips.unsw.edu.au/jw/circu
lar.htm

Therefore, equations related to circular
motion can be used to approximate the
velocity of objects in orbit

See notebook file for the derivation of
orbital velocity
C HANGE
IN ORBIT

What happens if Fg was to
change?

What happens if v changes?
K EPLER ’ S L AWS

Johannes Kepler a German
mathematician, astrologer and
astronomer determined 3 laws that
govern planetary motion

Though Kepler finally deduced the real
motion of the planets, he did so by
analyzing data gathered by another
scientist by the name of Tycho Brae
K EPLER ’ S F IRST L AW

Planetary orbits are elliptical,
with the sun at one focus of the
ellipse
K EPLER ’ S S ECOND L AW

The straight line
connecting the
planet and the
sun sweeps out
equal areas in
the same amount
of time
Kepler's Second Law
Interactive
K EPLER ’ S T HIRD L AW

The cube of the average radius , r, of a planet’s orbit is directly
proportional to the square of its period, T

Namely: r3 α T2

Therefore: r3 = CsT2

Where: Cs = constant of proportionality for the sun

Note that: Cs is based on the object that is creating the gravitational
field

Kepler's Third Law Interactive

See derivation of Cs in notebook file
U NDERSTANDING E SCAPE
E NERGY

Objects on planets are “bound”
to the planet in a situation very
similar to the following

Imagine being tied to a bungee
cord to another object, and the
only method of escape that you
have is to run as fast as you can in
order to try to “break” the cord
D ISCUSS
THE ENERGY

What must you do in order break
free? Discuss your energy
expenditure

What is the relationship between
your energy expenditure and the
distance between you and the
object?
Y OU ’ RE IN AN ENERGY DEBT

You can view the energy of this system in terms of
how much you “owe” the bungee cord in order to
get free

All the effort that you put in in order to break free
of the bungee cord doesn’t increase your speed –
it’s all put into stretching or trying to break the
cord

If the cord wasn’t there – you would be going at a
much faster speed for the same distance travelled!
E SCAPE
E
Radius of planet
r
ENERGY
H OW DO YOU GET AWAY FROM
THE BUNGEE CORD ?

An object exiting a gravitational field must do so by paying the
energy “debt” with kinetic energy

In order to escape the pull of the planet, the total kinetic
energy of the rocket must EQUAL OR EXCEED the energy debt
owed to the planet

ESCAPE VELOCITY refers to the minimum velocity required for
an object to just escape the gravitational pull of the planet,

ESCAPE ENERGY refers to the energy associated with the
kinetic energy required to repay the energy “debt”

See notebook file for derivation
T IED
UP IN DEBT

Any object that is “bound” to earth remains so
because its total energy does not exceed the
energy debt owed to the planet

Think about an orbiting satellite:

It remains in orbit around the earth (therefore, it
is still “tied” to the earth – how do we know?
What happens if it stops moving?)

But it also has a kinetic energy

See notebook file for derivation
T HE
E
DEBT IN ORBIT
r
r
Etotal
Ek
Eg
•Total energy for objects in orbit will equal to HALF
of the potential energy owed at that particular radius
TOTAL ENERGY OF OBJECTS IN A
GRAVITATIONAL FIELD

Therefore, total energy of any
object in a gravitational field is
therefore equal to the sum of its
gravitational potential energy and
the object’s kinetic energy

Therefore, there are 3 cases that
can be set up due to this
C ASE 1 – O BJECT JUST
ESCAPES : E T = 0



In this situation, the
kinetic energy is just
enough for the object
to escape
E
Ek
All the kinetic energy is
used to pay the energy
debt
What will the object’s
motion be like when it
escapes the field?
r
Eg
C ASE 2 – O BJECT ESCAPES
WITH V > 0: E T > 0

In this situation,
there is enough Ek
to pay the debt and
provide the object
with enough Ek to
continue onwards
with a constant
velocity as r ∞
E
Ek
r
Eg
C ASE 3 – B OUND OBJECT:
ET < 0


In this situation, there E
is enough Ek is not
great enough, so total
energy is still negative
Ek
Since there is still an
energy debt, the
object remains bound
to the planet and
cannot escape
r
Eg