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Momentum and Gravity
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Astronomy 309R  Spring 2009
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Motivation: A Review of Kepler’s Laws
(that Kepler inferred by studying Tycho Brahe’s measured orbit of Mars)
1.
The orbit of a planet is an ellipse with the sun at one of the foci.
2.
A line joining a planet and the sun sweeps out equal areas in
during equal intervals in time.
3.
(orbital period)
2
is proportional to (radius of the orbit)
3
http://xkcd.com
Implications of Kepler’s Laws
•
An implication of the 2
nd
Law: any single planet moves faster when it is
closer to the sun, and lower than when it is farther from the sun.
(Please
convince yourselves that you can reach this conclusion simply by
contemplating what is stated in the 2
nd
Law!)
•
An implication of the 3
rd
Law: planets farther from the sun move
slower
than
planets closer to the sun. Let the symbol “
∝
” mean “proportional to”.
So the
3
rd
Law can be written to state that: period
2
∝
radius
3
.
You can take a square
root of both sides, to obtain the proportionality: period
∝
radius
1.5
. The
velocity of a planet is equal to the circumference, which is 2
×
π
×
radius,
divided by the orbital period.
So the velocity is proportional to radius/period,
which is in turn proportional to radius/radius
1.5
, which nothing other than
radius
−
0.5
(note the minus sign!) that we can recognize as one divided by the
square root of the radius. In mathematical notation,
Clearly, the larger the radius, the smaller the velocity!
2/2/09
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View Full Document Some Questions Left Unanswered by Kepler
• Why do the planets remain in perpetual motion?
(Perhaps there is a deeper, more general physical
law that guarantees perpetual motion for planets
orbiting the sun.)
• What is the physical origin of the neat relation
period
2
∝
radius
3
?
(Perhaps there is a deeper,
more general physical law that implies Kepler’s
3
rd
Law, and from which many other conclusions,
distinct from Kepler’s laws, can be drawn.)
2/2/09
Astronomy 309R  Spring 2009
4
Concepts to Lean
• Momentum
• Isolated systems
• Conservation of momentum
• Newton’s laws of motion
• Newton’s law of gravitation
• Angular momentum
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Momentum
• Every body carries momentum associated with the motion
of the body.
The momentum of stationary bodies is equal
to zero.
• Momentum has magnitude (strength) and direction.
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This note was uploaded on 09/14/2009 for the course AST 309 taught by Professor Johnlacy during the Spring '08 term at University of Texas at Austin.
 Spring '08
 JohnLacy
 Astronomy

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