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{\Large ASTR430 Homework \#4 \\
Due Tuesday, April 3, 2001}
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1. Repeat Problem 2 from Homework \#3 for $D$, the range of a
projectile. Use the solution set to guide your work.  a) Make the
``flat Earth'' and constant gravity approximations and solve for the
maximum horizontal distance (range) that a projectile will travel in
terms of gravity $g$, angle of launch $\theta$, and launch velocity
$v$. \\ b) Use dimensional analysis to get an expression for $D$ using
the true $1/r^2$ gravity of the Earth. Neglect the Earth's spin and
treat the Earth as a perfect sphere.  Start by identifying four (or
five) variables that the solution will depend upon.  See if you can
form a dimensionless constant from the four variables and remember
that the most general solution is the one that is multiplied by the
most general dimensionless function.\\ c) Finally, solve for the
maximum range of a projectile (measured along the Earth's surface)
exactly and check your answer against part b).  Start by arguing that
trajectories will be segments of conic sections.  What conservation
laws can you use? After you get an answer, consider all of the
limiting cases that you can think of as tests for your answer.

2. Mass of the Oort cloud. In our Solar System, Jupiter formed at
about 6.0AU and was dragged inward to its current 5.2AU. This happened
as Jupiter interacted with icy planetesimals and threw them out of the
Solar System. Imagine that the planetesimals all were in eccentric
orbits at Jupiter's distance to begin with, and that they were thrown
out of the Solar System on parabolic trajectories. Use a conservation
law to estimate how much mass must have been ejected from the Solar
System to account for the changes in Jupiter's orbit.

3. Tidal forces between the Earth and the Moon are pushing the Moon's
orbit away from the Earth and slowing down the Earth's spin. What
happens in the end? \\ a) Calculate the Earth's spin angular momentum,
the Moon's spin angular momentum, and the Moon's orbital angular
momentum. Make the approximations that the lunar orbit is circular and
that the angular momenta are aligned.\\ b) Assume that only tidal
forces between the Earth and the Moon are operating and argue that the
total angular momentum of the Earth-Moon system is conserved. Is the
orbital energy conserved? Explain your reasoning.\\ c) In the end
state of orbital evolution, the Moon will be further away from the
Earth, than it is now, and the Earth will always keep one face toward
the Moon. This is the synchronous state (i.e. the ``month'' will equal
the ``day''). Use conservation of angular momentum to determine how
far the Moon will be from the Earth then, and how long the ``day''
will be. You may neglect the spin angular momentum of the Moon - use
your answer from a) to justify this. The final equation cannot be
solved analytically; try to get a solution with a numerical or
graphical method.\\d) That would be the end of it, but we've neglected
the effects of the Sun. Tides raised on Earth by the Sun slowly slow
Earth's rotation. How will the Moon's orbit react? (assume that the
Moon has already reached the synchronous state that you found in part c) -
because of the Sun's influence, the synchronous state actually will
occur with the Moon somewhat closer to the Earth). Describe what
happens to the Moon next using Angular Momentum and Energy arguments.



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