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[Inspired by some annos to 'Negative
Glider'. This one *is* an invention, though
of dubious feasibility and utility...]
HBman and VFFman are still arguing over
the results of their free-falling contest
(qv). A re-match is arranged, this time in
reverse. On the ground at the foot of the
tower block are two identical standard HB
catapaults, each capable of launching an
object vertically at 98m/s. The challenge
is to see who can reach the greatest
height, without any additional propulsion,
after this catapault launch.
"Aha!" cries the optimally-
streamlined VFFman - "You have no
chance of winning. If we were in a
vacuum, we would both reach an altitude
of 490 metres, decelerating to a standstill
due to gravity before falling back down.
As we are in air, we will both lose height
due to air resistance but, since I am
Optimally Streamlined, I will lose less than
you and will win."
"We shall see." says HB man, climbing into
his Super Slingshot Shot projectile. By
good fortune, VFFman happens to weigh
the same as HBman-plus-projectile. They
each sit poised on their respective
catapaults.
At the same moment, both catapaults fire,
projecting the two contestants upward at
98m/s. VFFman, optimally streamlined,
nevertheless experiences significant air
resistance, slowing his ascent a little more
than would gravity alone. HBman is
almost as well streamlined but not quite,
and begins to lag behind.
Suddenly, HBman opens a port in the nose
of his projectile, allowing the air to rush
past a variable-pitch turbine; the turbine
is connected to a heavy spherical mass,
which it sets spinning (and no, this isn't a
gyroscope trick!).
"Aha!" cries VFFman "I am sure to win!
HBman has foolishly traded some of his
upward speed for rotational energy of that
spherical mass! he is already falling
further behind me!"
Seconds later, both of their speeds have
dropped due to gravity and air resistance.
HBman cunningly alters the pitch of his
turbine, and now exploits it as a propellor,
driven by the momentum of the spinning
mass. As both contestants reach the apex
of their trajectories (and as HBman's
spherical mass finally slows to a
standstill), HBman glances down to see
that he has out-reached VFFman by just a
few inches. The winner!
"Cheat!" cries VFFman - "what you have
done violates energy conservation!"
"Au contraire", replies HBman. "I merely
exploited the fact that air resistance
increases more-than-linearly with speed.
At the bottom, when we were both moving
fast, I traded some of my vertical motion
for rotation of the spherical mass, losing
vertical speed faster than you did. Sure, I
lost some energy in the transfer, but not
as much as *you* lost through air
resistance at high speed. Then later, when
we had both slowed a lot, I simply
transferred the energy back from the
rotating mass into vertical motion. Again,
I lost energy in the transfer, but at these
slower speeds I didn't waste much energy
through air resistance. In effect"
concludes HBman "I simply used an energy
store to even-out my vertical speed,
thereby wasting less energy through the
disproportionately high velocities at the
start of the contest. QED."
"Incidentally", adds HBman "you'll note
that neither of us reached the theoretical
maximum height of 490m - otherwise we
would have violated all kinds of
conservation laws."
"Bugger." mutters VFFman. "Outbaked
again."
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Annotation:
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He does indeed suffer losses in
spinning up his mass, and again in
converting its spin back into lift. *BUT*
he is also spending less time travelling
at a high velocity, and more time
travelling at a slow velocity; this saves
his total losses due to air resistance (as
far as vertical motion goes), because air
resistance is more than proportional to
velocity. So, if his turbine/mass system
were lossless (which it isn't) he would
certainly win by evening-out his vertical
speed during the ascent. The only
question is whether the actual losses in
the turbine/mass system would
outweigh the gains through velocity-
evening. I don't know if there is an
absolute upper theoretical limit on
turbines in the same way that there is
on (say) a classical heat engine. |
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One more thing: at these velocities, it
probably wouldn't work. But at higher
launch velocities, the energy loss from
air-resistance at the start of the flight
will be **much** worse, and hence the
advantage of storing the kinetic energy
for use at later (slower) speeds will be
proportionately greater. For any given
turbine efficiency, there must be a
launch-velocity above which it would
give a net advantage.... |
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hmmm, I'm not so sure about this...So you're saying that the graph of wind-resistance vs velocity would be something akin to this
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There's a curve in there that suggests there is more resistance at higher speeds than lower ones btw.
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So, IF we use some of that force to speed up a turbine, we can release it later when there is less resistance to slow us down?
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Holy Perpetual Motion HalfBaked man!
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Sounds dodgy to me, more energy will be used to work the turbine than will be extracted from it - at whatever speed HB Man is going at. He will be slowed down more by the increased wind resistance as he is going fastest, and even though he's stored his energy, it will be a lossy conversion - both times.
So say his turbine is 99% efficient, he looses 1% of his upward intertia as he powers it, then later, when he tries to use the spinning turbine to power himself for another couple of seconds, he's going to loose another 1% for the conversion back from rotational to vertical motion. This means (at best) he will only be able to use 98% of the energy he collected earlier on. |
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He might be able to use the turbine to slow his descent, and even power a small christmas tree for a while by converting the turbine's energy into electricty. But it is a lot of bother to go to in order to achieve a festive, but out-of-season lighting effect. |
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Sorry, nice try, but unfortunately like for the negative glider you hit the second law of thermodynamics. |
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[Una] there are no outside forces in a glider system. Lift is generated by converting energy, in the case of a glider gravitational potential energy, in the case of other planes chemical potential energy. |
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[zen tom] - the thing about the air-
resistance curve is that it goes up
steeper than linearly. In other words, if
you double your speed, your wind-
resistance is *more* than doubled
(though not quite quadrupled, I think). |
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So, if you travel for (say) 10 seconds at
50m/s, you will lose less energy from
air resistance than if you travel for 10
seconds decelerating smoothly from
100m/s to 0m/s, even though your
*average* speed and *average* distance
is the same. |
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What it comes down to is that steady
speeds are more efficient (as regards
losses to air resistance) than changing
speeds. So, an energy-storage system
that can smooth-out the velocity will
give an advantage. In reality, losses in
the conversion will indeed count against
this advantage, but needn't completely
offset it. All depends on velocities and
efficiencies, rather than conservation
laws. |
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IIRC, wind resistance is proportional to the cube of velocity <Disclaimer: this is based on physics/maths lessons dating back to the mid-part of the second half of last century> |
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[Absinthe] - many thanks. I didn't think
it was as steep as cube, but if it is then
so much the better. But for this to work
in principle, it just needs to go more
than linear (square or 1.5th power
would be OK). |
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Absinthe, air resistance is proportional to the square of the velocity for a range of velocities where drag coefficient can be held constant. (due to viscous and sonic effects, drag coefficient is not constant at very low and very high speeds). |
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Sorry, this is still bad science. Even with a perfectly lossless turbine, you'd break even at best. |
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Freefall - thanks for the clarification on
air resistance. And I accept that a 'real'
turbine would be hopelessly lossy.
*But* I still argue that a lossless turbine
would make this work. A 'perfect'
turbine (by definition) would allow a
lossless exchange of vertical speed
(k.e.) into rotational energy and vice
versa, no? So, in theory the entire
ascent could be made at a constant
50m/s, rather than at a speed which
decreases linearly from 100m/s to 0m/
s (as it would for a normal projectile).
Now, since air resistance increases as
the square of speed, the total losses
during the journey at 50m/s will be less
than the total losses during the journey
that starts at 100m/s and ends at 0m/s.
Hence, less total energy is wasted in air
resistance and the projectile must rise
higher (since there's nowhere for energy
to go ultimately except by being
dissipated by air resistance). |
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If you're driving a car, you waste less
energy in wind resistance if you drive
at a constant 50mph than if you drive
for the same time (and distance)
acclerating steadily from 0 to 100mph,
even though the average speed is the
same. This is the equivalent argument
turned vertically; we're just using the
turbine to spread out the imparted
kinetic energy of the launch so that
velocity becomes more nearly constant
during the ascent. |
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As I said, I am sure that at any realistic
velocity, and with a turbine of any
practicable efficiency, you'd lose out.
But with a perfect turbine you'd win,
getting closer to (though never
reaching) the height which you would
have reached without air resistance. No
violation of energy conservation, just
minimising air-losses by keeping closer
to a uniform velocity. |
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Very interesting idea [Basepair], and great annos. |
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This reminds me of a news video clip about a vertical rollercoaster. The car went straight up a tower, coasting, and the owner was bragging about how it was in freefall. The newsman had a tennis ball which did *not* float for the camera. The owner just ignored it and the newsman didn't argue with him. It took me a while to figure out that the car's wheels were acting as flywheels, keeping it moving up and then slowing its descent. |
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HBman's projectile is similar to that rollercoaster, except that the flywheel is spun-up after launch. And the spherical shape is a rather bad one for an energy-storage device. I'm going to take the principle off to a rotorcraft design, and see if I can add anything to this. |
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VFFman's projectile has a definite upper limit to its trajectory, streamline it how you will--purely ballistic. HBman may be said to be flying, even if rather poorly, by supporting his craft with moving air. |
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For example, and for less moving parts, HBman could used a winged craft, which upon vertical launch used its wings to do an immediate sweeping turn to the horizontal. Ideally, it would then be gliding at 98m/s and using its wings for lift and its momentum forward for energy to climb. Practically, it would lose energy on the turn, or be gliding at 98m/s, which is what HBman was planning to avoid. But a video of his possible paths, run backwards, could be compared to the negative glider, and show that as [Basepair] says in the idea, HBman can't go any higher than a ballistic path. |
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My best design for HBman is a rotorcraft, based upon various flying toys and too little sleep. The rotor is mounted on a frictionless bearing, with two perfectly-designed blades incorporating tip-weights and pitch control.
After clearing the launcher, the rotor blades begin spinning up, and convert some of the forward motion to rotational energy. The blades then change pitch and shape to follow the ideal curve of lifting energy while giving the rotational energy back. (I know this is the original idea, but I've done rotorcraft in real life {such as it is} and I think this would be more efficient than a turbine and a spherical flywheel.)
So, with perfectly-streamlined blades (ignoring the fact that I'm now putting spinning blades into the airstream) that convert all lift to energy and all energy to lift, we start the video camera and launch this puppy.
When we run the video backwards, we see a roto-glider begin a slow drop, with its blades pitched for maximum lift, beginning to spin-up the craft. It passes through an curve of speed of rotation versus descent, then, when within only a few feet of the ground, the blades apparently malfunction horribly, converting all the built-up rotation into a mad smash into the ground at 98m/s.
VFFman's craft, seen in the background, drops from the same height, falling faster, but reaches terminal velocity due to air resistance, and has nothing to cause a final burst of speed, and hits the ground at a slower speed, but gets there first. |
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Okay, that was odd, but to me it says that energy storage would be better, provided it was done ideally. But if rotor blades can have perfect efficiency, so can streamlining, which makes the whole thing pretty iffy. |
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Ideally, a perfectly designed craft would be a cylinder with a variable-geometry inlet and exit nozzle. Upon launch, the air hitting the front of the craft would channel down the frictionless inlet to a storage tank, slowing, compressing and heating, and slowing the craft. The compressed air would later be released out of the frictionless exit nozzle, boosting the craft as it expands and cools. Both nozzles would be varying geometry like mad.
Looked at that way, I see HBman's craft as simply a rather odd way of going about making a pressure-recovery aft section. |
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I'm going to say that HBman would win for any launch speed greater than VFFman's terminal velocity, provided he has ideal energy to lift conversion.
Sounds like a great contest for Junkyard Wars. |
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I think HB man could win. There are some red herrings in here which might cloud thinking about this. |
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Let us imagine that VFF and HB have frictionless go carts on a level surface. The gocarts have huge coiled springs on board, each containing the same amount of energy. VFF discharges his spring all at once, pushing off against a rock, and goes zooming away. HB discharges his spring little by little through a series of gears, powering his car along at 10 kph. VFF is sitting still in his cart somewhere out there at the end of his wild ride. Will HB pass him before his spring is exhausted? |
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It depends on how efficiently the HB cart can turn spring energy into motion. Essentially, the turbinepowered ball in the original example does the same thing - capturing catapult spring energy by way of air resistance. An efficient vehicle could win it for HB man. |
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[Baconbrain] - Many, many thanks - you
have analysed this a lot more rigorously
than I had, and your anno is very
helpful. |
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Re "But if rotor blades can have perfect
efficiency, so can streamlining, which
makes the whole thing pretty iffy." - I
agree, which does indeed make the
whole thing a little moot. On the other
hand, given slightly-less-than-perfect
turbine efficiencies and slightly-less-
than-perfect streamlining, my gut
feeling is that, *at high enough launch
velocities* the turbine system would
still win, simply because the penalties
of air resistance would have more
impact on the not-quite-perfect
streamlining. However, I can see
converse arguments and I'm not sure
which is right. |
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Re the use of a gliding path, I agree that
we can never get higher than the in-
vacuo-ballistic path under any
circumstances, and I also agree that the
gliding velocity would have to be high
after an efficient turn, thereby not
saving air-resistance losses. So, gliding
doesn't seem to help. |
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I also agree regarding the design of the
turbines - I can imagine that the
spherical mass is not the best flywheel. |
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Your pressure-recovery system sounds
better than the turbine system. I was
thinking last night about having internal
sprung weights which would lag behind
the body of the craft on launch and
then spring back during the later stages
of flight to return the energy. But then I
figured that these would be cheating, if
the launch velocity of the catapault is
constant, since they'd be drawing extra
catapault energy to extend the springs.
But your air-compression idea sounds
like it could be done without cheating
(as long as no nozzles open until the
craft leaves the catapault; otherwise
you'd again be drawing extra launch
energy from the catapault, which would
be sort of cheating). |
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[bungston] - many thanks for the
clarifying anno. You're right in that this
is effectively a vertical equivalent of a
'go cart' experiment, and your
illustration makes the point that
smoothing-out the speed will be more
efficient overall. |
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Anyway, as a 'thought experiment' this
has been grand! Many thanks to all
annotators! (Wanders off to try to sell
idea to Burt Rutan and/or former
Supergun designers......) |
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Great physics. I like it. Better than all these "backward time whirlpool bubble" ideas we get nowadays. |
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Several dual stage versions are possible. For example, you could build a cannon which is lunched upwards. Compressed air is stored, which is later used to fire a projectile. |
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The body of the cannon would reach a reduced height, but the projectile could reach a greater height than that achievable even by frictionless projectile motion. Yes, it's cheating, but still, it introduces a slight greyness to //we can never get higher than the in- vacuo-ballistic path under any circumstances//. And no extra energy is stolen from the catapult. |
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What if the projectile were also a (smaller) cannon, etc.? |
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Basepair's internal sprung mass would be a stealthier way of cheating. It would have to spring back _before_ the projectile left the catapult, so that the internal mass is moving faster than 98m/s, and therefore the RMS velocity of the contraption exceeds 98m/s on launch. |
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If this were not catapults, but spring powered helicopters, there is probably a way to calculate the ideal rotor speed to generate maximum height. It seems obvious to me that not all rotor speeds will attain the same height. VFFmans catapult launch corresponds to an arbitary rotor speed in the spring powered helicopter - not necessarily the best one. |
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