This may be spectacularly inefficient, or
then again it may not. Frankly, I have no
idea.

So.

Imagine a sleek titanium cylinder sitting
upright on the ground. Its base is flat;
the
top is funnel-shaped, funneling down to
a
tube within the cylinder. This tube is a
gun.

Upon
activation, a charge fires a
projectile
out of the large cylinder. As it rises up
some thousand feet or so, this projectile
trails behind it a light, strong cable,
harpoonesquely.

When the projectile reaches the limits of
the tether, two things happen. First, the
projectile itself springs open, releasing a
large but simple round parachute.
Second,
a winch in the cylinder is activated, and
starts to wind the cable in as fast as it
can
go. Quickly, the parachute inflates,
affording considerable air resistance. In
response, the cylinder starts to rise
upward, winching itself toward the
parachute. Although the chute is
descending slowly, the cylinder is
climbing faster, and the two meet
eachother at about 800 feet.

The winch continues to wind the
parachute
in, drawing it back into the funnel and
collapsing it in the process. As it does
so,
a deftly ingenious mechanism swiftly
concertinas the parachute and its lines
and
re-stows them in the shell of the
projectile. All this time, of course, the
contraption will be falling but, in the
second or so that it takes to re-pack the
chute, it only falls a few tens of feed.

As soon as the chute is re-packed in the
projectile, another charge fires it
skyward
again, and the cycle re-commences. Of
course, firing the projectile upwards
propels the launcher downwards but,
thanks to their relative masses, the recoil
is not too violent and little height is lost.

At each cycle, then, the launcher winches
itself up 800ft, then falls back maybe
100ft. The result is a noisy but
impressive
reciprocating walk up to any chosen
altitude.

I was pondering shooting guns in the air, as I often do. With shame, I realized that my math was inadequate to determine how high a bullet would go given a muzzle velocity. I thought perhaps it would involve calculus, applying gravitational acceleration (here slowing) to a projectile with an initial velocity. The shame, the shame. My face is as red as my bikini briefs.

But here Max has opened the door to an honest question. Given a projectile with a desired apogee of 1000 feet, and disregarding drag from air or any dangly appurtenances, what must the initial muzzle velocity be? Show your math.

The maths is easy, and doesn't involve
calculus if you ignore drag. As always,
there is a shortcut.

As it leaves the muzzle, the bullet has
velocity V and mass M; it has kinetic
energy of M.V.V/2 . At its peak, it has
zero velocity but has converted its
kinetic energy into gravitational
potential energy, given by MGH (where
G = acceleration due to gravity, and H =
height).

Since we are looking at the complete
conversion of kinetic energy into
gravitational potential energy, we have:

M.V.V/2 = MGH

and hence:

V.V/2 = GH

and, since G = 10m/s/s:

V.V = 20H

or:

V = Sqrt (20H)

or roughly

V = 4.5.SqrtH

So, if you want an altitude of 1000
metres, then:

V = 4.5 x Sqrt(1000) or

V = 4.5 x 31 or

V =140m/s

Note that the square root relationship
means that extra velocity buys you
*lots* of extra height. This makes
sense, because a faster bullet not only
travels further during the initial, fast
part of its flight, but also takes longer
to slow to a standstill, so you get a
double bonus.

I was going to suggest you could probably do this with levers and a rotary engine, but then the whole suggestion devolved into a paddle-wheel helicopter, and from there, to a regular helicopter, getting less impressive as I went.

//As soon as the chute is re-packed in the projectile, another charge fires it skyward again// and the launcher is shot towards the ground at a high rate of speed...?

The other interesting thing about this idea
is that the main body of the device will
(unlike a conventional rocket) be subject
to relatively low rates of acceleration,
which could make it a more comfortable
ride. If the harpoony bit can take very high
rates of acceleration you can take
advantage of high explosives to launch it.

You are going to need a succession of ever larger parachutes to gain the same drag in the rarified upper atmosphere. Unless your mechanism is becoming lighter.

Your projectile is dragging weed.

You are going to need to keep your firing cylinder pointing up, once the parachute is winched into position.

This is similar in many ways to the Schermuly PAC (Parachute and cable) low level air defence system developed in conjunction with D.M.W.D. in the early 1940's. That used a vertically launched rocket and parachute, trailing a cable, to provide an alternative to barrage balloons. <link>

A quick calculation indicates that the mass of the system will be too great for "flight". But [+] for a HalfBaked idea.

//and the launcher is shot towards the
ground at a high rate of speed...?//
and

//drag enhancers that deploy at the
second stage firing, to minimise the
recoil//

Both true, but not excessively. If the
launcher weighs 50x the projectile, then
the recoil velocity (and hence distance
lost) will be only 1/50th that of the
projectile.

//You are going to need to keep your
firing cylinder pointing up// Good
point.

//two legs, i mean wings, for the
vertical flap walk // Undoubtedly more
appealing.

//the mass of the system will be too
great for "flight"//

I'm not so sure. Consider a typical
round parachute as a reference point.
When carrying a human (say 100kg) it
has a descent rate on the order of a
couple of metres per second.

So, we need a system which weighs
about 100kg all in, can fire the chute
ballistically to (say) 1000ft, and then
winch itself up the wire fairly swiftly. I
don't think this would be impossible.

Of course, you could scale the whole
thing, and larger versions may be more
efficient (?), but at least for a 100kg
system it should work.

//The maths is easy, and doesn't involve calculus if you ignore drag// I had to do some calculations a while back on 40mm AA rounds. Although at firing, they underwent something like 25000g, as soon as they were out of the barrel, they started decellerating at something like 10g, even in near-horizontal flight. Ignore drag at you peril.

Calculus is my middle name. Physics is my first, and witty repartee is a distant third.

Ignoring the effects of drag when dealing with velocities of around 140m/s is like ignoring the effects of heat when mining the sun.

Taking an assumed projectile of 2kg mass (gleened from numbers bandied around here) and about the size of a metric pot roast, with a somewhat streamlined drag coeffecient of 0.5, and a muzzle velocity of 140m/s, we get an apogee of 461.5m.

As this is considerably less than the 1000m predicted when ignoring drag, we can see the perils of such an assumption.

Any request for alternate shapes, sizes and muzzle velocities?

One could dispose of the chute after the cylinder was maximally reeled in. It will never deploy propoerly a second time anyway. Chutes have to be packed just so.

The cylinder would contain a stacked series of chutes, made of thin film. On maximal reeling in, fire the new chute through the first. When I say dispose of the chute, of course I mean make it explode. The chute film is highly flammable and will ignite from the exhaust of the next fired chute, burning in a harmless cloud of flame as the projectile passes through it.