MOTION IN SPACE

In 1543 Copernicus published On the Revolutions of the Heavenly Spheres in which he proposed that Earth and the other planets orbit the sun in perfect circles

The astronomer Tycho Brahe made many precise observations of the planets and the stars, but some of his data did not agree with the Capernican model.

KEPLER’S LAWS

Johannes Kepler, an astronomer, worked for many years to reconcile Copernican theroy with Brahe’s data.

His analysis led to three laws of planetary motionThese laws were developed a

generation before Newton’s law of universal gravitation

FIRST LAW- Each planet travels in an elliptical orbit around the sun, an the sun is at one of the focal pointsThe first law states that the planet’s orbits

are ellipses rather than circlesHe came about this lab while trying to

make sense of Mars’ orbitHe experimented with 70 different circles

and finally realized an ellipse with the sun at a focal point fit the data perfectly

THE THREE LAWS

SECOND LAW- An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time interval

“Law of equal areas”If the time it takes a planet to travel the arc

of section c (tc), is equal the time is takes to travel the arc in section b (tb), then the area Ac is equal to the area Ab

Planets travel faster when they are closer to the sun

THIRD LAW- The square of a planet’s orbital period (T2) is proportional to the cube of the average distance (r3) between the planet and the sun, or T2α r3

Relates the orbital periods and distances or one planet to those of another planetThe orbital period (T) is the time it takes to finish

one full revolution

This law also applies to satellites orbiting the Earth

323

2

31

22

21 or , rT

r

r

T

T

According to Newton’s Third law, T2

α r3, the constant of proportionality between the two turns out to be 4π2/Gm

Where m is the mass of the central object

Thus, Kepler’s Third law can also be stated:

32

2 4r

GmT

Solving for orbital period:

SPEED OF AN OBJECT IN CIRCULAR ORBIT:

In both cases m is the mass of the central object

PERIOD AND ORBITAL SPEED EQUATIONS

Gm

rT

3

2

r

mGvt

During Magellan’s fifth orbit around Venus, it traveled at a mean altitude of 361 km. If the orbit had been circular, what would Magellan’s period and speed have been?

Given: r1=361 km=3.61 x 105 mT=? v=?

Sample problem

Radius of venus : r2=6.05 x 106 mMass of venus: m=4.87x1024 kgr=r1+r2=6.41 x 106 mT=5.66 x103 sVt = 7.12 x 103 m/s

Solution

Gm

rT

3

2r

mGvt

This is not the absence of gravity

It is the absence of a support force

An elevator explains this nicely- The sensation of weight is equal to the force that you exert against a supporting floor

When the floor accelerates up or down, your weight seems to vary

Weightlessness