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Path: utzoo!watmath!clyde!floyd!harpo!ihnp4!zehntel!hplabs!sri-unix!David.Smith@cmu-cs-ius.arpa
From: David.Smith@cmu-cs-ius.arpa
Newsgroups: net.space
Subject: Moon bullets
Message-ID: <12070@sri-arpa.UUCP>
Date: Sat, 7-Apr-84 15:58:36 EST
Article-I.D.: sri-arpa.12070
Posted: Sat Apr  7 15:58:36 1984
Date-Received: Fri, 13-Apr-84 06:00:44 EST
Lines: 37

Problem:  How close to horizontal must you fire one of those high-powered
rifle bullets for it to stay in orbit?

Let's treat the moon as a perfectly smooth sphere of 2160 miles diameter,
and assume that our marksman fires from a height of 6 feet.

The horizon is 0.083 degrees below the "horizontal," and is located 1.57
miles away.  If one fires below the horizon, the bullet will surely hit the
moon, whatever its velocity.  If one fires above that with infinite
velocity, it will miss the moon.  0.083 degrees corresponds to an elevation
error of 5.2 inches in 100 yards.

But the fastest the bullet can be fired into a return orbit is just below
parabolic velocity.  To prevent a bullet with parabolic speed from dropping
more than six feet, the maximum depression angle is 0.0588 degrees, which
corresponds to an aiming error of 3.7 inches in 100 yards.  At this
angle, the bullet will impact 2.2 miles away.  There had better not be a
six-foot hill within 4.4 miles of the firing point.

By symmetry, this maximum depression angle is also the maximum elevation
angle.

What if the moon is littered with three-foot hills?  Then you had better not
fire at an elevation or depression of more than 0.04 degrees, corresponding
to an error of 2.6 inches in 100 yards.

If the bullet has less than parabolic velocity, these margins dwindle,
until they reach zero at circular velocity.  (If the bullet goes around at a
constant six feet, it will run into highlands sometime.)  Rougher terrain
also cuts into the margins.  Then there is the moon's rotation, which over a
month's time is bound to put something in the way of that six foot periapsis.

As I recall, Apollo spacecraft circled the moon in about two hours.  That
puts circular velocity at about 3400 mph, and parabolic velocity at 4800
mph.  Mighty fast rifle bullets.

				David Smith