Gravity: Newsletter Challenge (11/14/06)

Trevor A. HART, Trevor Hunt, was my (unpaid) ADC and good neighbour, RAF Techie, and perfectly invaluable during the 1999, Serbian Operation. He knew all there was to know about Rocket Ordnance, and I knew practically 'Nudda'.

The trebuchet weight which is used to fire the stone
would remain relative to the wait of the stone
This would mean it should fire the same distance regardless of the pull of gravity
but less gravity means less resistance from the air so it might go further
assuming there is no vacuum if there was it will travel further still

For a long time I've read these posts; agreed with some points, chuckled at others, but kept my comments to myself. However, today I feel compelled to comment. The question states,

" Your daughter is working on a physics problem for homework. The problem asks her to find ways to improve the firing range of a trebuchet. She's having an argument with your son, who is insisting that you could just move the trebuchet to a planet with lower gravity than the Earth. The lower gravity would allow the projectile to fly farther. Your daughter is saying that's wrong. You step in to settle the argument. What's your answer?"

So we're looking at range with respect to gravity of a gravity powered device. Negating mechanical friction due to modern bearings... this could be an improvement?..., (and air resistance) there are simple high school physics equations at work here. Don't overcomplicate things or you'll drive your kids away from physics and engineering. Just give them the tools they need and let them settle it themselves... The'll learn more.

Lastly, I have to agree with europium. Don't you think that perhaps before you posted a technical reply on a website subscribed to by technical minded people that you would understand the question and more importantly the physics behind it and heres the biggy.... not make false assumptions or blatant errors in equations!

To those of you that know what your talking about. Keep up the interesting posts, I enjoy the discussions immensly. I believe it was Mark Twain who said, "Better to keep your mouth closed and be thought a fool than to open it and remove all doubt"

Someone may have stated this already, but if the reduced gravity is acting on the projectile making it travel further what is it doing to the counter weight used to turn the Trebuchet arm to hurl the projectile, this will also be subject to less gravitational pull thus the turning force of the main arm will be reduced. Will this not mean a loss of efficiency, taking you back to square one.

The simple answer would be to add wheels I don't know what I am talking about, I pinched this of wikipedia, there are some interesting trebuchet sites out there

Extract from Wickipedia:-

"A trebuchet can increase its efficiency (and hence either range or payload) by allowing the counterweight to take the straightest possible downward path. This maximises the transfer of the counterweight's potential energy to the projectile rather than to stressing the frame. Mounting the counterweight on a pivot straightens the path of its fall, increasing its effectiveness. A fixed counterweight trebuchet in particular can therefore be made more efficient by the addition of wheels to allow the frame to move freely back and forth. This also allowed the trebuchet to fire farther.

The addition of wheels also makes the trebuchet more stable as part of the forward momentum of the falling counterweight is transferred to the forward motion of the trebuchet instead of a tilting action of the vertical frame, possibly tipping over of the machine or severely damaging the structure. The velocity of the trebuchet frame is added to that of the item being thrown, increasing its velocity and range by up to 33 percent. The wheeled trebuchet can effectively employ a fixed counterweight, mounted to the short end of the throwing arm, rather than the pendulum weight described above. The weights were usually stones and rubble, since lead was far too expensive and could be used for better purposes in a siege."

The projectiles used could be anything from rocks to dead animals, bee hives or captured soldiers or Engineers"

Keep it simple lads its High School Physics Homework

Hi Guest, Back in those olden days. Warfare was governed by certain etiquettes, in the event of captured soldiers being lobbed over the walls, a Herald might be sent to 'have a quiet word. Herald derives from the old German, Dez Herez Zu Walten, and became shortened to Dez Herevolt..etc. The Herald might address the commander of the siege thus "Dear Sir, we much appreciate the generous gift of Honey, and our dogs are very grateful for the meat, but please refrain from returning our soldiers in such a fashion, it is undignified. They should be treated with respect." if the herald was sent away without a satisfactory answer, then the monks would procure prostitutes who were known to be incubating smallpox or plague. End of siege. If the Herald was harmed, as sometimes happened, then the home town of the offending general would be ploughed over, when he returned home, not a stone would be standing upon another, his kith and kin would have vanished.

Postscript:- Occasionally, a Monarch might not agree to allowing a favoured General's home town, being ploughed over. So be it, "And a pox to you and your court" would be the not so diplomatic response, Smallpox can with care and medication be survived. It is terribly disfiguring and then make-up needs to be applied with a trowell. White Lace 'Ruffs' hide the neck. WMDs go back a long way. Heralds were known in the earliest dynasties of Ancient Egypt. Where warfare is concerned, Heralds had no allegiances, except to the 'Creator'. Even if a Herald had learnt that his own Monarch was to be ambushed and taken captive, he would not breath a word. Except after plea for a lenient ransome. Alexander the Great thought he might defeat Kazakhstan, His troops were sent packing in swift order, with an invitation to return and face certain death. Two Roman Emperors lost their entire armies to ventures in the middle east. The second Roman Emperor was chained to the ground as a foot stool. for everyone to mock. History....as we are seldom taught in school.

The boy is partially correct.

In an environment with lower gravity the projectile will fly farther. There are though another two parameters affecting the range: the initial velocity with which the projectile leaves the trebuchet (the higher the velocity the farthest the range) and the atitude angle that is the angle with the horizontal plane with which the projectile leaves the trebuchet. For maximum range this angle should be set at 45 degrees.

Guest you stated in post #70

"In an environment with lower gravity the projectile will fly farther."

Since I went to the trouble of doing the mathematics to prove it wouldn't could you please elaborate on you claim?

Hi Masu - relax. The point about a trebuchet being a projector that is driven by a counterweight has now been made several times, so initially this contribution irritated me too. But there's not a lot we can do about people who don't bother to read or understand the thread - unless we start to filter everything - and that's not really the objective, I feel.

Thanks for posting those equations, Masu, but I have to confess that for a minute I thought I needed new glasses.

-e

Sorry about that the equations didn't come out that good. I defiantly need to develop a better way to insert equations. Please feel free to enlighten me, all suggestions will be happily received.

In a word: beer.

-e

Masu wrote: "I defiantly need to develop a better way to insert equations."

I get pretty defiant sometimes when it comes to inserting equations. It's still a problem my therapist is trying to help me work through as it tends to make blind dates somewhat awkward for the three of us. Thank heavens the great dane knows how to dial 911.

-e

Masu, you wrote: "Please feel free to enlighten me, all suggestions will be happily received."

I found a very roundabout way: create equation in Word or TEX (or find it somewhere), use a PDF writer type application to print it as a PDF. I then copy and paste from there to a graphics package where I have choice in saving it as a JPEG. When inserted into CR4 using the 'camera', it looks OK (Jorrie's Blog shows the effect).

But what a schlep! I also wish someone can give a better way!

Here is a thought on creating mathematical images.

First create and store a master image file using the MS Paint utility program. You should only need to do this one time. This creates images as bitmaps that can be saved as .bmp, .jpg, .gif., or several other image formats. Paint has a text creation tool, so you could select various fonts to use from its tool bar, including the standard UniversalMath BT, a True Type font. This font produces Greek letters for the Latin equivalents on the keyboard, as well as special mathematical operators and other symbolsfrom the numbers and other non-letter keys. If you got good with this font you would not even need to use a master image file, but create the characters as you needed them. The text toolbar lets you select highlighted text and make it bold, underlined, or italic, as well as changing fonts and size. The drawback is, you cannot edit the created text, since it is now just a bunch of pixels, but you can create a new graphic with new, correct text and sybols and just paste it over the old one.

Paint allows you to quickly and easily copy and move any section of the graphics image on the screen, so groups of symbols you repeat or use frequently could be saved in a corner and erased later when you are done.

Of course lining everything up is "by eye" and depends to some extent on having a good "mouse hand", but with practice this method works fairly well.

Then merely do a "Save As" to get the file name and format you desire for easy reference and loading as a .jpg or other image file.

This saves the two steps you used to create the PDF and then convert the PDF to JPEG. Any mathematical image you can collect by cut-and-paste can also be pasted into PAINT.

PAINT may not be the most powerful graphics program, but it is universally available and relatively easy to use, and the current version which I have (5.1) is a huge improvement over previous versions.

Hi Stl_Engineer thank you for your suggestion. Unfortunately that's pretty much what I do now. The only difference is I use Microsoft Equation to generate the formulae and then paste them into Microsoft Paint. The problem appears to be when I try and use the picture function in CR4. It changes the size of the image and it looses resolution. All the equations you see earlier are BMP images from Paint. I tried using a larger image but no matter what I did the CR4 editor always reduced it to the same size and the result was the low resolution you have seen.

Have you tried using the larger font sizes in the text toolbar?

Here is one of my favorite "integrals" in size 14 (first example) and 36 (second example) :

I did cheat a little. The text in the second example was in Arial bold. But look at the integral symbol. It was these same font (Symath) but the size 36 has a higher resolution than the pixelated size 14 version.

Interesting response. The equations that I have generated though look fine while that are in Paint but when I Use the picture button in CR4 it all screws up. The other thing I noticed is that you can only insert a limited number of images that's why I have grouped the equations together rather than having them with the text that refers to them. I do everything using Microsoft Word and Equation all within a Word document then at the end I just pick up the text and paste it in the CR4 editor. It would be good if the equations came with it. It sounds a bit convoluted but by doing it this way it gives you all the editing tools as well as the spell checker.

Hi Masu, I was put on to a package by Mathworld

Checkout:- http://mathworld.wolfram.com/

There are some nifty software packages out there, but boy was I in for a shock, mathematical symbols seem to have bred like fibbonacci rabbits! I could recognise about 3% of them, but after a Wee Dram or two, about 5% ish? Alastair

Perhaps you are being too ambtious about the format. I think it would be good enough to write your entire set equations in the same style as equations 1-3. It wouldn't be ideal, but it is quick to do, it works, and it's clear.
N.B. if you get fed up with "Sqrt(expression)" you can always use √(expression), (expression)^0.5 or (expression)^0.5 .

The most serious limitation I have found (so far) relates to symbols that need subscripts and superscripts with the same lateral position. For example, ∫^r_s is probably clear, but rather inelegant.

Physicist I agree that I could do it that way bit I find it gets very confusing when you end up with complex equations like

S_m/S_e = Sqrt(g_m/g_e) x Sqrt(2h/g_m/2h/g_e)

that don't really convey what is going on. Its not immediately obvious that the something like this

V_2m/V_2e = Sqrt(2g_mM₂(M₁R₁-M₂R₂)/Sqrt(2g_eM₂(M₁R₁-M₂R₂)

is actually equal to

V_2m/V_2e = Sqrt(g_m/g_e)

as it is when you write it out correctly. Mind you with the way they turned out earlier you have a valid point since it's impossible to read.

Yes, it's not as transparent - but it doesn't matter much for the posts I've seen from you - because you (helpfully) provide the necessary steps anyway.

Being like Alaskan, I too normally set in the bleachers and enjoy the comments and "proofs" offered. I recall from early physics classes the first thing we are taught is 1."there is not air" and 2."there is no friction". With these "truths" embedded and held forever inviolate, the father would have to side with the daughter and say that the gravitational differences of two different plants would have zero effect on the ultimate range of the trebuchet. Simplification can be a good thing for us dads.

Thank you UMR 70, I thought to Google up "There is not air" and came to some good educational websites, but then came to a SENSATIONALY RELEVANT Website,

Check out :- http://www.damninteresting.com/?p=359

Damn Interesting indeed, "Gravity Powered Aircraft". Mmmmm? Cool or What?

Alastair said: "Damn Interesting indeed, "Gravity Powered Aircraft". Mmmmm? Cool or What?"

Cool and workable as well... maybe impractical?

It is really a combination of "gravity powered" (through buoyancy) and "solar powered" (through temperature differences at altitudes), with perhaps some small conventional energy input.

Thank you Jorrie, (P.S. I know what I want in my Christmas Stocking, but I will have to buy it myself probably. A book on cosmology....by some engineer or other? Dorothy Nelson at our local bookshop will place the order.) Not as impractical as sending a trebuchet to another planet.

Gravity powered aircraft, as I see it would needs be very large. 'buoyancy' is the working principle. float up to ten miles altitude, retract wings and glide down with a 'prop' storing energy for powering compressors etc. later in the next buoyancy phase.

To maintain the 'Trebuchet' thread, assume our treb is on wheels, which according to wikipedia gives up to 33% extra throwing distance, I wonder if a light structure would improve on a heavy? i.e. would balsa wood be better that steel girders? How about an inflateable 'Trebuchet' one you can take down to the seaside, and fill the weight with sand? I thought that Gravity powered aircraft might benefit from Professor Caracticus Potts' 'Helium Filled Struts' tubes of woven Kevlar reinforced non gas permeable membrane, inflated with compressed Helium, net weight 'Nudda'? reasonably stiff in the circumstances, that might even be an advantage?

Filling the trebuchet swing arm or for that matter any of it with helium wouldn't affect it at all. Yes you would reduce its weight but it still has a mass and it's the mass that is the problem. Keep in mind mass and weight are two different things.

Sorry Masu, That's the problem with straying 'off thread' Helium filled struts were for lighter than air 'aircraft' re; 'Gravity propelled aircraft' It was the Jesuits who thought no castle or city would be safe from aerial bombardment, hot air balloons were supressed science about four centuries ago. I just wondered if one could make an 'inflateable trebuchet' perhaps for hurling water-balloons at the sea side? Something that might make these two noisey kids arguing about low gravity planets, get back down to earth?....remind them of holidays to come, as a destraction.(psychology)

Masu, I note with interest that 'Trebuchets' are the 'Office Gadget' of the moment. Desk Top Trebuchets for consigning junk mail to the 'round fileing cabinet' in the corner. Scrunched up into a neat ball naturally. Very therapeutic I should immagine

Now if we could just get an electronic one that worked on SPAM it would be extremely satisfying.

Professor Caracticus Potts, taking inspiration from his 'pyramid power' thinking cap, A conical paper hat with a large 'D' was thinking Inflateable Trebuchets, Sand Castle Demolition, for the purpose of.... That throwing arm would need to be 'stiff' if it was made to have a 'conical' shape like the 'thinking cap' it would hurl much better?

I am awarding 5***** Stars for entertainment on this thread, I have learnt a lot, and also laughed till my sides hurt! Thanks everyone. and especially GlobalSpec CR4 Team.

Since the trebuchet is likely powered by a counterweight method, I concur with the daughter. If the trebuchet is powered by a counterweight method, and with all things being equal, the launch distance of the projectile in a lesser gravity environment would be reduced.

If the trebuchet were powered by an elastic kenetic energy source (springs, etc.), then the launch distance of the projectile in a lesser gravity environment would be increased.

I would first say to them "If you had the skills to get yourselves to another planet, I don't think you would be worrying about building a better trebuchet!" However, I think if you kept the same general ideas on this planet, you might have a good science fair experiment. Fire the treb at sea level, and fire it on top of Pike's Peak.

The Earth's gravitational pull decreases with elevation (distance from the center of mass), but the air also becomes thinner (and doesn't a baseball travel further in the Rockies' balpark?).

Yes! That is the problem, trying to separate the two effects gravity on mass and atmospheric drag. That is why someone creates a "thought experiment" like this. If it were practical to create the real experiment, then of course, that is what we would do!

I know, let's build nano-trebs and fire them by remote control inside an evacuated bell jar!

If the ratio of the weight powering the trebuchet and the missile was the same to the first approximation the range would be the same.

But in the lower gravity envirenment assuming the limit to the powering weight was set by the strength of the structure a larger weight could be used increasing the range

Agreed. So I would say to my son "this is a man you want on your side".

Hi Guest, Find the biggest elevator lift, and use one of those 'Toy' Tebuchets. A 'velcro' ball as the missile would stick to the carpet, and assist measurement. Going down and up would certainly alter the 'Gravity'. especially if fired at just the right moment. A 'Spring Ballance' with a heavy weight could determine the best moment. i.e. use a Department Store elevator. Buy the spring weighing scale for the bathroom, the tape measure for the toolbox and the toy treb and velcro ball , from the Toy Department.

since a trebuchet relies on gravity as part of its propulsion mechanism (counterweight), there should be *only* atmospheric effect variation.

The projectile will fly farther, but only as an indirect result of the lower gravity.
Since the trebuchet is a gravity powered device, the lower gravity affect on the
projectile once launched is offset by the lower gravity effect on the weight that
powers the device. This actually results in a lower initial velocity.


The slight gain to be had in distance has to do with the drag on the projective caused
by the atmosphere. A planet with a lower gravity will have a less dense atmosphere,
leading to less drag on the projectile. Less drag will let the device fly farther.

I also believe a lower initial velocity will result in less drag because drag is affected
by the square of the velocity.

Do you have anything original to add? All that you have said, correct or incorrect, has been said before. Don't you read the previous postings?

A trebuchet is powered by the dropping of a heavy mass. The energy applied to the projectile is equal to M("power" mass)*g(planet's gravity)*H(height of "power" mass for trebuchet)* (whatever the efficiency of the trebuchet is). We will call the efficiency "e". We should assume that both trebuchets fire at 45 degrees for optimum distance. Ignoring air resistance, the projectile flies in a parabolic path. Because of this, the projectile's maximum height can be found by measuring the distance it travels (assuming flat ground), and its maximum distance can also be found by the maximum height it reaches. We know this because we have a "zero" at the launch site as well as at the landing zone, and a maximum at the maximum height it reaches, and we also know that the slope at the first zero is equal to one. So let's find the maximum height reached! The velocity leaving the trebuchet can be found from the energy applied by the trebuchet, which is equal to M* g*H*e. At 45 degrees, the maximum height reached can be found by the equation Energy = m(projectile)*g*h(max height of projectile)*sin(45). Putting these together, we have: M*g*H*e = m*g*h*sin(45) Solving for h, we get: h = (M*H*e)/(m*sin(45)) The maximum height reached is independent of gravity, since the trebuchet is powered by gravity. We can then use our assumption of no air resistance to say that gravity will not affect the trebuchet.

However, with air resistance, and a similar atmosphere, the exit velocity of the trebuchet can be found from M*g*H*e = ½ * m * V^2, or exit velocity equals (2*M*g*H*e/m)^.5

More gravity equals higher exit velocity, which means more initial wind resistance, and a less efficient "fling" so that the total height would not be as far with the higher gravity. This is assuming that the atmosphere is similar.

As far as atmospheres go, we could argue all day that less gravity equals less density, but who is to say that an alien planet has the same gasses in their atmosphere.

This brings me to my final answer, which is this:

Any earth-rock given the chance to be flung through an atmosphere would undoubtedly go farther on an alien world, regardless of what is used to fling it or how strong the gravity is. This is due to the fact that a boost skyward is a perfect way for the rock to explore, and take in some nice birds-eye views of the mysterious new planet. Also the rock will have ample opportunity to scope out places to meet hot alien rock chicks (or dudes if the rock being flung is female... or if it swings that way).

When in doubt, go with the obvious.

Quote

but if the reduced gravity is acting on the projectile making it travel further

OOPS I was wrong my appologies.....

By Guest Omega:

As the challenge is written, the daughter appears to be correct, as any reduction of projectile weight which might permit the boulder's "flight" range to be extended would be offset by the reduced weight of the impelling trebuchet counterbalance. But, consider…when the challenge is stated using the syntax that was probably intended, wherein it might be restated: …lower gravity would permit the objective of improving firing range to be achieved.

In that case, before being interrupted, what might the son be arguing, for which the daughter's physics course has evidently not prepared her? Possibly: that she has failed to take into account the weight of the trebuchet itself—especially the non-stationary part? Possibly the persuasion which the son hoped to impart derives from his own lessons in Automobile Mechanics 101? In which course it was taught, that given constant engine output torque, weight could be added to a vehicle without proportional reduction in vehicle speed (in vehicle carrying capacity)…provided!…provided that the added weight is suspended weight. Perhaps the son was attempting further to illustrate: that the same, say, 200 kilograms added as suspended weight would not affect vehicle velocity nearly so much as adding the weight to the vehicle wheels? Perhaps he might have added that this…lies behind the "drive" to increase vehicle performance by using wheels of decreased weight? Had father not intervened, might the son have also been able to persuade his sister that the trebuchet, particularly its hurling beam, is essentially a wheel? And that the counterweight represents a counterpart to a motor vehicle's engine output? Just as if it was imparting torque—increasingly—at the beam's pivot point? And, that the weight of the projectile represents the counterpart of a roadway's resistance—equivalent to a vehicle's weight—to a motor vehicle's engine torque? Thus, of the vehicle's velocity relative to its weight? And, as such, if increasing the weight of a trebuchet's "Wheel" (its unsuspended weight relative to the projectile) would diminish the potential projectile throwing velocity for a given counterweight force, then does it not follow that any decrease in the throwing beam's weight—such as would be attained on a "lighter" planet—would contrariwise increase the throwing arm's potential velocity independently of the weight of either, projectile or counterweight? And that, the decrease of trebuchet weight, particularly its throwing arm's weight, ought to confer an advantage that will permit the machine to hurl the projectile faster, hence farther, on the "light" planet than on the "heavy" planet?

Hi Guest Omega, We assume the 'Trebs' on both planets are identical, weight might be lower, but 'MASS' is constant, lower weight is often confused with lower mass, that will not be changed in my view? The torque inertia will be the same, that might even act as a 'disadvantage' on the low gravity planet? (time for applied maths)

She is right, as the gravity force that provides the energy for the catapult is reduced in the same way as the energy demanded by the parabolic range. Both in a linear way.

Most respondents will direct their analysis to the fact that gravity drives both the initial acceleration supplied to the projectile as well as the acceleration that limits its range, and will conclude that there is no net benefit. Less obvious is that low gravity means a less dense atmosphere, hence less wind resistance, hence longer range.

The son may be displaying an early appreciation for fluid mechanics.

This was already covered in some detail. Suffice it to say that Venus is lighter than earth and has a very much denser atmosphere.

It is a logical error to pose that the atmospheric resistance would be the same. Lower gravity = lower atmospheric pressure = lower resistance. Why pretend otherwise?

A tougher question would be: Does the resistance change in a proportional relationship to the change in gravity, or would it be non-linear? Without research, I'm thinking non-linear, thus an even greater distance improvement than the proportional gravitational differences.

Additionally, less gravity would reduce friction on the mechanism = greater distance.

Dear Guest, may we submit further 'Logic' ..."More Atmosphere = More Atmospheric Pressure"? i.e. Venus= lower gravity= more atmosphere= more atmospheric pressure

If both planets had no atmosphere, I agree that the firing range would be the same. However, with an atmosphere, the planet with the least gravity would probably have the thinnest atmosphere, hence the least drag and the greatest firing range.

This was covered many times in the trail. In principle, atmosphere should depend as much on history, and on distance from heat sources and larger gravitational bodies as it does on size. Venus is a good illustration of this. Also, improvement due to reduced atmosphere must be greater than other losses. Contribution #105 (if the gravity is lower, you can use use a heavier weight to power the trebuchet) was probably the best in support of the son.

I recommend care regarding atmosphere - if the atmosphere is too thin, the moisture in the wood will boil, and the trebuchet will explode (as will the operator).

OK enough.

Being the dedicated punkin CHUNCKER ( one which propels pumpkins with a trebuchet)

we tested one in a zero gravity chamber ( SMALL SCALE TREB USING PING PONG BALLS FILLED WITH WATER) and guess what. IT DID PROPEL AT A HIGHER SPEED UNTIL CONTACTING A WALL.

How did you create the zero gravity chamber, line the bottom with anti-matter?

I seriously doubt even the most dedicated "punkin CHUNCKER" would pony up for a ride on the "vomit comet" just to test a miniature trebuchet!

Even NASA isn't that crazy!

Even if you did such a thing, the treb would not fire under zero gravity because there is nothing to move the counterweight. A trebuchet will NOT work in zero gravity!

just turn the trubuchet on its side... fix the slings so they must stay in line with the original gravity... and you can simulate near zero gravity. You can also simulate a much larger gravity by putting the treb in a large centrifuge.

Chris

Learn from Ze Wikipedia:- Ve assume zat both trebuchets are ze same. i.e. ze components have ze zame mass. zere vil be a lezzer forze acting on ze low gravity planet, but ziz forze vill have to onercome ze zame 'inertia' rotating about ze axis. zerefore ve conclude zat ze projectile vill receive lezz acceleration 'proportionately' on ze lower gravity planet. (Professor Caracticus Potts)

Scalar moment of inertia Definition

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

where

m is its mass, and r is its perpendicular distance from the axis of rotation.

The moment of inertia is additive so, for a rigid body consisting of N point masses m_i with distances r_i to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia

Generalizing to a solid body described by a continuous mass-density function , the moment of inertia for rotating about a known axis can be calculated by integrating the moments of the point masses relative to the rotation axis

where

V is the volume region of the object, r is the distance from the axis of rotation, m is mass, v is volume, ρ is the pointwise density function of the object, and x, y, z are the Cartesian coordinates.

I would say that it go the same distance becuase gravity pulls on all things at a constant rate, therefore it would pull less on the counterweight and the projectile causing it to fly relatively the same distance as it would on planet Earth.