Tuesday, December 11, 2007

Thought Experiment: Orbiting Cannonballs

A baby puts everything in its mouth because its mouth is one of its most effective sensory organs.   For adults, "seeing is believing", because we (or at least those of us who are sighted) get most of our information via our eyes.  But what do we do when we want to observe something beyond our experience—something not accessible to the five external senses?  We imagine, of course.  But when that imagination is tempered by logic and care, it really becomes an eye on an internal world.

Physicists call it the Thought Experiment. Actually, since it was popularized by Albert Einstein, whose native tongue was German, it is usually referred to by its German name Gedankenexperiment, which literally translated is ..."thought experiment".
  [Aside: I love the song "Die Gedanken Sind Frei", which I learned of from the Unitarian Universalist hymnal.]

This diagram is due to one the most influential scientists in history, Sir Isaac Newton.  (More about him and my little intersection with him later.)  It is a diagram designed to explain how objects can be put into orbit.  It was drawn more than 250 years before Sputnik, and nearly 100 years before the first balloon flight.  It is still, in my opinion, the clearest way of explaining how satellites orbit.

A cannon sits atop a mountain, which by the looks of it is around 500 miles tall, which is about 100 times the height of Mt. Everest.  It shoots cannonballs that fall back to the Earth in the usual parabolic path.  Each successive cannonball is shot faster so that it goes farther than the last.  Eventually, when a cannonball's trajectory curves more slowly than the orb of Earth, it falls forever without running into the Earth.  In other words, it goes into orbit around the Earth.
The only way Newton could "see" this was in his mind.  This thought experiment allowed Newton to extrapolate from the world of his everyday sense to things on the scale of the Earth or even the solar system.  With this simple diagram, he showed how things orbit.  The moon orbits the Earth because it falls, due to gravity, in a gentle curve around and around the Earth.

But it is important to look at a diagram like this with care.  It is not enough to glance with your inner eye.  You have to look carefully.  Confession: When I looked at the diagram, I assumed that by fine tuning the speed of the cannonball, you could get it to hit anywhere around the Earth, say where the A or the B are.  But they are not potential landing sites.  I should have thought about it more.  Then I came across this Java applet which lets  you simulate the trajectory for any speed you wish.  I soon realized that once the cannonball is fast enough to make it past the South pole, G, it would make it all the way around.  You can't get it to hit near A or B (neglecting friction).   Try it.   Find the largest speed for which the cannonball still hits the Earth.  Then see what happens 1 mile/hr more than that.  Like many other physical situations, there is a symmetry that can tell you what is going on.  If the cannonball can make to the South Pole, then it traces a path on the other side of the Earth that is a mirror image of the path it took getting there, and hits the cannon from behind.  The whole path is an ellipse with the closest point at the South Pole.
[confidence level: established, my qualifications: trained]


yajeev said...

Lots of fun. I look forward to more Gedankenexperiment.

mars said...

Greetings EyesOpen

Thanks for your thoughts of such wide array. Look forward to your next volley.

RE: Die Gedanken sind frei

I rather much enjoy parts of the following translation, which contains an elegant passage lost in other translations:

Sie rauschen forbei
wie naechtliche schatten

"They pass like a whisper,
like an evening shadow"


Or from the Leonard Cohen version:
"They fly along like nightly treasures"


Thoughts are free
Who can ever know them?
They pass like a whisper,
like an evening shadow,
nobody can know them,
no hunter can kill them,
no it’s been agreed:
thoughts are free
I think what I want
and think what I like,
but all in silence,
as must be,
my wishes, my hopes,
nobody can stop them,
so it’s been agreed:
thoughts are free.
Lock me in the
darkest dungeon, too,
but all’s in vain
because my thoughts
will break down the barriers
pass thru’ the wall.
Thoughts are free
I now want to give up
love forever
and I don’t want to ever torment myself with irrational fear
each one can laugh
and joke in his heart;
and think;
thoughts are free
I love wine,
but most of all, my girl,
most of all,
I love her,
I am never alone
with a glass of wine
and my girl, beside me
thoughts are free.

Die Gedanken sind frei
wer kann sie errathen?
Sie rauschen forbei
wie naechtliche schatten,
ein mensch kann sie wissen
kein Jaeger ersschussen
es bleibet dabei:
die Gedanken sind frei.
Ich denke das Ich will
und was mich begluecket
doch alles in der Still
und vie es sich schicket
mein Wunsch und Begehren
kann niemand fervehren
es bleibt dabei
die Gedanken sind frei.
Sperret mich auch ein
in finsteren Kerker
so sin es doch nur vergebliche Werke
den meine Gedanken
zerreisen die Schranken
und Mauern entzwei
die Gedanken sind frei.
Nun will Ich auf immer
der Liebe entsagen
und will nicht mehr
mit Grillen so plagen
Man kann ja im Herzen
stets lachen und scherzen
und denken dabei
die Gedanken sind frei.
Ich liebe den Wein
mein Maedchen vor allen
die tut mir allein
am besten gefallen
Ich bin nicht allein
bei einem glas Wein
mein Maedchen dabei
die Gedanken sind frei.

(modified by moi to fix some obvious grammatical errors)