Gravity: Newsletter Challenge (11/14/06)

I think in the case of a 100% efficient trebuchet, there will be no difference, given that the planet's atmosphere has the same density than Earth's. In the case of a practical, real life trebuchet and a real life planet, I will take my son's side!

Hi there,

If we assume that a typical missile fired from such a weapon (a big rock!) would have a fairly significant drag coefficient then the density of the atmosphere on the respective planets would thus effect the trajectories.

For this to be correct, I'm assuming the counter weight would not reach the same velocity as the fired projectile and therefore any drag acting on this part of the system would be less than that acting on the missile.

I think the plant with the lower gravity would have a thinner atmospheric density (?) which would therefore enable the rock to go further.

If the missile were shaped like a dart, this difference would be less.

I think the daughter is correct.

Ian

opps i meant to say the son is correct!

-assuming of course all other things being equal (ie atmospheric composition).

-density of the atmosphere and thus drag (at any given altitude from the surface) would be effected by gravity.

You guys want to have your cake and eat it too! You want to assume an Earth-like atmosphere, but NOT an Earth-like gravity! And it appears that, for Jorrie at least, no one else seems to be concerned about it, there is an assumption that gravity will have a significant effect on efficiency, while a well-designed modern Trebuchet should make mechanical friction insignificant.

I think the original question was designed to determine whether or not people thought reduced gravity would have a direct effect on distance, disregarding the minor effects of atmospheric drag and mechanical friction.

Hi STL, I reckon your: "I think the original question was designed to determine whether or not people thought reduced gravity would have a direct effect on distance, disregarding the minor effects of atmospheric drag and mechanical friction" hits it pretty well 'home'.

I don't think kids are supposed to consider all these engineering complications. We should rather let them think about plain, simple, basic truths. The fact that the likes of us (engineers) will never in our (or their) lifetimes agree on the 'correct' answer, should not corrupt them. Or worse, put them off engineering completely!

In the meantime (don't tell the kids), let's have more fun!

Jorrie,

I don't mind the engineers who argue the sniggling little points , but understand and correctly state the main principles in effect , or those who give a qualified answer or even a guess, admitting what they do not know , as much as those who call themselves engineers, or who should know better, and are NOT kids, but give answers that are way off-base, while trying to sound like they are THE authority on the subject! Even worse are the idiot's who call a correct answer "wrong", but never admit THEIR mistake!

My other pet peeve is those who give a "me-too" answer that has already been stated, and re-stated, over and over again, without really adding anything to the discussion, like all the posters on the "Lightning" thread (after the first half dozen anyway) had the same basic "Faraday cage" answer.

-me thinks someone needs a holiday and soon...

You say that there will be no difference if the atmosphere is the same. I think this would be true if the drag coefficient ("CD") was unchanged. But in practice drag coefficient is not constant, being high at low velocity and gradually reducing to the 'nominal' value once the velocity is well into the turbulent regime. Although the velocity is probably significantly into the turbulent regime, the effect will probably not have bottomed out - so, at least in theory, the range should reduce under low-gravity conditions if the atmosphere is constant. A lighter atmosphere would change this - but it is not universally true that lighter planets have thinner atmospheres - Mars and Mercury do, but Venus has a much denser atmosphere than Earth. Therefore, the situation regarding the effect of atmosphere appears to be indeterminate.

If there is any tension in the construction of the trebuchet, that too will tend to maintain a gravity-independent friction, which would reduce the range. The resonant frequencies of the arm and of other parts of the system will presumably not change significantly, so the effect will depend on the relationship between the time between triggering and projection and the vibration quarter?-periods for the arm, body, and sling-tension. If the trebuchet was already well designed for earthly use, we should expect the increased firing time under reduced gravity to degrade all of these.

Without many more constraints, I'm sitting firmly on the fence as to what the effect would be (more haemorrhoids) - which I suppose means that I support the daughter in her view that the sons prediction is baseless.

Physicist wrote "... - which I suppose means that I support the daughter in her view that the sons prediction is baseless"

I agree. With all those devils in the details, support the daughter, because her project (of improving the trebuchet) is at least easy, while the son's suggestion is outrageously impractical…

I believe everyone is missing the point. With less gravity the trebuchet will have less thrust due to less pull on the counter weight.

The starting point for Jorrie and followers was that, neglecting air resistance and losses, the energy of the projectile is proportional to gravity, and the range of the projectile is (first-order) proportional to energy/gravity. So, in an ideal world, range would be independent of gravity. Therefore, the changes in the range depend on the non-idealities - losses in the trebuchet mechanism, air resistance...

Exactly. I agree completely

I have to weigh in on your side in this argument and say that a tebuchet operates on the basis of gravity. If you change the gravity it is going to proportionally change the distance. If gravity is half, then the distance will be half

The point is that range is proportional to kinetic-energy divided by gravity, and kinetic energy is proportional to gravity, so they cancel.

Try the simplified case where the projectile starts and finishes at the same height, and there are no losses:

flight_time = 2*vertical_velocity/gravity

range = horizontal_velocity*flight time

Initial kinetic energy = constant*gravity=mass*speed*speed/2

Horizontal_velocity=speed*cos(projection_angle)

Vertical_velocity=speed*sin(projection_angle)

Substituting:

range = speed*cos(projection_angle)*2*speed*sin(projection_angle)/gravity
= cos(projection_angle)*sin(projection_angle)*2*speed^2/gravity
= cos(projection_angle)*sin(projection_angle)*4*consant*gravity/gravity
= cos(projection_angle)*sin(projection_angle)*4*consant

You can see that gravity cancels out exactly. Showing that, neglecting losses, the range is independent of gravity. This was also shown algebraically by masu (post 55).

Right on !! Even though I do not have an engineering degree this is the thought that

entered my mind as I read other responses. Less gravity would propuce less thrust,

however a less dense atmosphere would be less friction so I guess everything would be more or less the same as here on Earth ?

Jorrie,

Usually you are so good with explanations. Instead you just made a judgement. Let me take a crack and see if you agree.

A trebuchet works on a principle of conservation of energy. A load is lifted up under gravity and given some potential energy based on mass x height. When the load is released it turns a wheel or lever, that accelerates a projectile payload along a curvilinear path, turning potential energy of the gravity load into kinetic energy of the projectile. Since the projectile is also working against gravity, if it is loaded and fired on the same planet, and disregarding atmospheric effects, as you noted, it should follow basically the same ballistic path. I believe it could even be loaded on one planet, say Earth, and fired on another with the same atmosphere (Planet X), and there would still be no difference, since the load would be in the same relative position regardless of where it was loaded. That is, the potential energy of the load is converted to kinetic energy of the projectile, which then moves upward, working against the same gravity that established how much potential energy was stored in the system. Yes, the load releases less energy on a planet with lower gravity, but it also takes less energy to fly just as far, as long as, like you say, the Trebuchet is 100% efficient (all potential energy is converted to projectile kinetic energy and none into heat, vibration, or work against the Trebuchet itself, e.g. moving it along the ground).

Now on the other hand, what if we used a similar device that works on a different principle, the Catapult! Would a 10 kilogram stone travel the same distance on Planet X as on Earth? Assume that the catapult is wound up the same amount every time.

Hi STL, I'm a lazy writer - usually wait for someone like yourself, Europium or, especially, Hero to do the explaining!

I think you are basically right with the trebuchet, but you have not given the full 'real life' answer!

As far as the catapult is concerned, that's a quite different kettle of fish, with tensional/torsional potential energy instead of gravitational potential energy.

But let's hear what other posters have to say.

OK, Jorrie, I'll bite. When you say "full 'real life' answer" are you talking about gravitational effects or atmospheric effects, or something else? In what way?

Physical motion equations, like the nice - clean - summary equations that cameo listed, have lots of "nearly insignificant" factors left out and only the variables that are significant to determine the 'motion' are generally used.

These not-significantly-large-enough factors include, off the top of my head; the effect of initial static as well as kinetic friction at the rotation connection point, effects on the projectile of the release from the sling, the stretch effects during acceleration of the sling material - which change based on ambient conditions such as temperature and humidity, the effect of choice of material and the resulting properties of the frame during firing - which react to ambient conditions as well as gravity, the dampening in a real system that takes place when the design itself (height, arm length, sling length, firing/release point, mass of counterweight) are not 100% efficient and some parameters overshoot their peak contributions to the projectile at slightly different points, whether the Trebuchet frame is perfectly anchored, how much holy-water soaked into the projectile when blessed before firing, and of course the aerodynamic effects previously mentioned along with probably quite a few that Jorrie could throw in that I would not likely think of even if I took a longer coffee break here...etc.......

Check out:- http://www.the-office.com/bedtime-story/hopi-windgod.htm

Bedtime-story....Don't mock that 'Holy Water' Kid's prayers are always answered. 'Hopi Yaponcha' the Wind God may just oblige with a favourable gust of wind? The question was, 'How to improve the performance' Bedtime story Sorted as well!

Hi STL, you said: "When you say "full 'real life' answer" are you talking about gravitational effects or atmospheric effects, or something else?"

I guess I talk about all of them. Is static and dynamic friction in the contraption dependant on the gravity of the planet, or not? Does very light bodies maintain atmospheres in general? What else does real life throw at the engineer?

Without being chauvinistic, I think the son, supporting the lighter planet, has a better chance of becoming a successful engineer than the daughter! Or am I wrong?

hello folks,

if you want to get rid of the drag coefficent and the gravity, then you would have to fire it in space. tere you would get the max. distance out of it...

tom

Tom,

A trebuchet is dependent on gravity for its energy. You may be thinking of a simple wind-up catapult.

Don't fret. Others have made the same mistake.

Here's my opinion.

If the gravity were near zero, and the atmosphere non-existent, the trajectory of the thrown object would be a near-straight line, and its speed would be the final velocity upon leaving the trebuchet.

Therefore, given all other factors being equal except gravity, the object will definitely travel farther in lower gravity, as gravity bends the trajectory and thereby shortens the distance to impact. (assuming the trebuchet is a good design, and doesn't throw the rock directly at the ground.)

chris

But then again, if gravity were zero, then there won't be any potential energy Ep=mgh=0 to be converted to kinetic energy Ek=0.5mv^2. So velocity will be zero and your projectile will go nowhere.

I don't think so. The energy is stored in 'spring' form, in the trubuchet, and is then applied to the projectile upon triggering. Gravity does not provide the potential energy to the object in the first place. A big hairy middle age(d) warrior with a metal cone shaped helmet has to wind it up!

You just described a catapult not a trebuchet. A trebuchet is a simple counterweighted beam so if the two masses remain constant and assuming no losses then no matter how big the gravity field is, the projectile will have the same energy upon leaving the thrower. It is the net difference in moments about the hub that will fling the projectile.

OOPS...

Thankyou

How did the catapult get in this discusion? A trebuchet and a catapult are two different machines.

Also, lets correctly differentiate between weight (an effect of gravity) and mass (which doesn't change in relationship to gravity). Mass*Velocity=Force, the projectile is going to travel roughly the same distance on any planet as long as you don't exceed escape velocity on that planet.

I mentioned the catapult in post #2, just to see who knew the difference and it was interesting.

Also, you have agreed with me and several others that range would not be increased simply because gravity is less, but Jorrie makes a good point about atmospheric drag and friction, just unfortunately worded in the idea that a lighter planet must have less atmospheric drag. Please see the comments about the example of Venus.

Incredible... never heard so much non sense.. the daughter is correct ...go girl power.. !

Mass * Velocity doesnt equal force... check your units..

last time I checked.. mg = f (mass) (gravity(or acceleration)) = Force (Newtons or whatever)

mv = p which is momentum... I've been our engr college for 20 yrs and havent forgot any of this s**t.

What exactly do you think the big hairy middle age warrior with a metal cone shaped helmet is winding up? Counter weight! On a low gravity planet would this action still be the same as on earth? I don't think that the counter weight will start the initial volocity as it would on earth so i'm not sure the rock would even leave the sling. i would go with the daughter.

hi guest, Ep do not equal zero because the gravity will be less not zero ,and that ia not our matter, but i think that the throw will be less because of the Ep will reduce asumming that the air resistance is neglected.

_______________________________________________________________________

"There is two infinte things:The universe and the human stupidity,and I am not sure about the universe"Albert Einstien

Welcome to the thread Petro, have a good read, some of the comments cover just about everything there is to be said.......unless you know of something omitted, then please tell us. You are correct to state that a planet 'MUST' have 'SOME' gravity.

The daghter would be right. The son's theory would be right if the same amount of gravity would applied to the loads on both trebuchets. But, the trebuchets are on diffent planets. The son would be right if you increased the load on tebuchet 'X'. Then the trebuchets would be different.

But then I would ask them why did they nuke it so much. Make a bigger trebuchet. Der.

Or put infinate gravity on the load and very low gravity on the projectile. Then you would get very good results. but that would be changing the laws of physics.

Maybe I am not clear on the strict definition of a trebuchet but it seems to me the distinction is in the fact that it has a sling as a mechanical impedance transformer as an adjunct to the basic catapult mechanism that could be powered by gravity or an elastic or even other method. Perhaps there is a "classical" definition that establishes it to be one way but I have seen modern trebuchets powered by compressed air. So, I feel the question is incomplete. However, in the case of a gravity powered unit, I will go with Jorrie's answer. Powered by a source that is not gravity driven, of course it will go further.

Trebuchet converts potential energy to kinetic energy s.t.

(Δm)gh=0.5mv²

v²=2(Δm)gh/m

So, initial velocity is dependant on the gravitational acceleration. Looking at the projectile motion:

vsinα=gt_flight/2

t_flight=2vsinα/g

Flight range is;

x=t_flightvcosα=2v²sinαcosα/g

=2·(2(Δm)gh/m)sinαcosα/g

=4(Δm)hsinαcosα/m=2(sin2α)(Δm)h/m

Hence, the gravitational acceleration cancels out for the trebuchet, it does not appear in the range formula. But, the picture would be totally different for spring driven systems.

Certainly, air drag not considered.

From the range formula:

x=t_flightvcosα=2v²sinαcosα/g

=v²sin2α/g

we see the well known fact that the maximum range for a projectile with a fixed initial velocity is obtained at exactly 45 degrees, since sin2α is maximized at 45 degrees.

Another interesting challenge would be what happens if the projectile is located higher than the ground. What do you think would happen to the 45 degrees, should it be smaller, higher or still 45 degrees for maximum range?

hi there,

I think that to acheive the maximum range the missile should be traveling just about vertically by the time it reaches the ground.

i.e. all of it's horizontal component of velocity is 'used up' against air friction.

Thus the angle of launch would be dependant upon the hieght of the launch pad.

with a 45deg launch angle the projectile will be travelling at 45 deg when it passes the launch horizon (same height as gun) and will continue on it's parabola untill it reaches vertical and therefore goes no further -with respect to horizontal range that is.

If the launch pad is high enough then the ideal launch angle would be horizontal such that the max energy is transfered into the horizonal vector of the missile's trajectory.

Ian

Hello Ian,

In all of my comments, I disregard air drag, since it makes things so complicated. The air resistance is in the negative direction of the velocity of the projectile and can be proportional to itself of its square or both, I don't think any analytic solution can be found if it was included.

It is true that if the missile is located higher than the ground, we should sort of invest more on the horizontal component of the velocity vector, since the flight time is already increased sufficiently by the increased height of the pad. I know it is a little bit vague to say, but consider the launch pad is sufficiently high. Then the vertical component of the velocity will bring just an incremental increase in flight time, but the horizontal component will increase it proportionally (x=t_flightv_horizontal). So, if the launch pad were at the infinity, the projectile had to be purely horizontal for max range.

Although I won't give formulation here, the formula for range in this case is;

x=vcosα/g·(vsinα+sqrt(v²sinα²+2gd))

I couldn't find a closed solution for alpha, but you should set dx/da=0, and find a for any d. It has such kind of a relation.

hi cameo,

can we ignore gravity also -to simplify even further ; )

Hi Ian,

Ignoring air drag for low speed projectiles is common practice, and accurate enough. It also allows people to say some quantitative things on the motion. Without it, we would be left with heuristic comments as "I think that to acheive the maximum range the missile should be traveling just about vertically by the time it reaches the ground."

Also, otherwise, there can be no single answer. Air drag is related with velocity, shape, surface and anything you can think of.

Make things as simple as possible, but not simpler. A. Einstein

In the absence of air drag, maximum range would require start and finish trajectories at about 45-degrees to vertical (taking the range to be very much greater than the difference between the start and finish altitudes).

45 is the optimum angle for same altitude.

My question is, what angle for different start and finish altitudes, you mean still 45?

Uphill = start steeper (than 45-degrees), end shallower.
Downhill = start shallower (but always upwards), end steeper.
The trajectory will be parabolic, and these angles depend only on the gradient between the start and finish points.

N.B. this is neglecting losses, and taking the horizontal component of velocity to be constant (sufficiently-large-planet approximation). If there are air losses, the projection angle will need to be nearer horizontal, and the final descent will be steeper.

I'm putting my trebuchet on a planet with such a low gravity that it can not sustain an atmosphere. Therefore I no longer have to assume zero air drag, there will be exactly zero air drag! This will yield the furthest projectile.

hi, but what about the gravity affecting on the projectile, after leaving the trebuchet?

Hi Petro, and welcome to CR4, We have had a great time with this thread, and if you read the posts, we all concluded that on a low gravity planet, the structure could support a more 'massive' but clearly not 'heavier' counter weight. That would be an advantage. If say ordinary stone was used here on planet earth, up there Wolfram Ore from 'Tung-stone' (heavy stone) could be used instead.

However, that might be cheating. Besides an atmosphere can also assist the flight, if you fire with a strong wind, for instance.

I have a question. The trebuchet relies on gravity to fling a projectile on a trajectory. I am assuming that the projectile and trebuchet are the same for both experiments.

So, the gravity is different, but the mass is not! It still requires the same PE to get the ball moving. Does the lower gravity cancel out the mass issue? That is, although the gravity is lower you apply less energy, but less energy is required to go the same distance.

If that is true, then the answer would seem to hinge on air resistance. You can't depend on the air resistance to be lower on a smaller planet. Venus proves that! Also, drag depends on velocity and I don't think the relationship is linear.

"but the mass is not!"

Sorry, Anonymous, but your unfortunate terminology belies the validity of the thrust of your comment. The MASS is EXACTLY the same, it is the "weight" of the Counter-weight (and the projectile) that changes with lower gravity on a planet of lower mass.

However, if I understand the intuitive point you are trying to make, a properly balanced Trebuchet will operate (will not 'stall' upon release of the firing catch) regardless of the gravity, so long as we are talking about a planet, not some theoretical rock with 'near zero' gravity, which takes you out of the linear response range at near-zero PE, in a "real" system, which is the reason for the 100% efficient assumption.

And yes, as Jorrie also briefly noted in the first post, it is the atmospheric density (drag on the projectile) that would make the difference, and which would be lower on a 'real' planet with lower gravity.

Therefore the son is correct, assuming "all things are equal" except the gravity.

Sandman said "And yes, as Jorrie also briefly noted in the first post, it is the atmospheric density (drag on the projectile) that would make the difference, and which would be lower on a 'real' planet with lower gravity."

I am not sure if that was the point Jorrie was trying to make, but he should speak for himself, which I have invited him to do!

I agree that atmospheric drag has an effect, but I think A. Hero also has a point, not all "real" planets of lower gravity have less drag. Venus, as he said, is a good example. Venus has a slightly lower gravity (8.87 m/s² ) on its surface than Earth (9.81 m/s² ), yet it has a much "thicker" atmosphere, much higher pressure and so more atmospheric drag. Scientist explain why this is possible by what is called the Goldilocks phenomenon. So one cannot count on a "real" planet with lower gravity to have lower atmospheric drag, so lower gravity alone will not necessarily allow the projectile to fly farther, and the son is wrong. Now he could of course find a 'real' planet with lower gravity and lower atmospheric drag, but then he should have said THAT, not just lower gravity!

Since Jorrie said that my answer (disregarding atmospheric effects and energy loss in the Trebuchet) was basically correct, but was not for a "full 'real life' answer" I am still waiting to hear his answer. He also said he would agree with the son, "In the case of a practical, real life trebuchet and a real life planet". Well, Venus is a "real life planet" and I don't see how the thicker atmosphere of Venus would give a "real life" Trebuchet any more range. If Jorrie meant that a "real life" answer was for a "normal" planet with lower surface gravity, which should have a lower atmospheric drag, then he may be right, but then what is "normal"? Does that mean "Earth-like"? If so, then that should be included as a qualification, not just "lower gravity". I still believe that Venus is the example that disproves the theory. The son is wrong if he believes that reduced gravity alone will create increased range.

Will the mechanical efficiency of a Trebuchet change just because it is on a planet with lower gravity? Yes, since there will be less friction due to the weight of the Trebuchet components on its main bearing, efficiency will be higher and more energy will be converted to kinetic energy rather than being wasted as friction generated heat or work, giving the machine more range. How significant is this? Hard to say. For a medieval Trebuchet made of wood and iron it probably would be fairly siginificant. For a modern Trebuchet using low-friction bearings, not so much.

surely the high temperatures on venus would burn our poor little trebuchet before the first shot was fired?

perhaps a point for the sun?

Ian,

If our Trebuchet is going to travel from the Earth to Venus, it will have to meet NASA standards, so of course its design will withstand the heat of the Venusian atmosphere (460ºC/680ºF). It will likely be made of a hight-temp steel alloy for its structural parts. Ball bearings will also be high temperature rated, however normal petroleum based lubricants would be replaced by high-temp dry lubes like Molybdenum Disulfide.

Our counterweight will be made up of surplus Space Shuttle tiles, stacked up and cemented together with ceramic bonding material, while our projectile will be a perfectly round and smooth ceramic ball which will burst open on impact releasing a brightly colored marking powder which could be picked up by powerful imaging systems overhead, therefore giving minimum atmospheric drag effects and allowing a quick and accurate measurement of the range by our robotic aircraft, a kind of UAV that has already been proposed for exploration of Venus.

Talk about flights of fancy!

well said, -however i think if we had the ability to build such a 'fancy-rock-slinger' it beggers the question -why would we?

surely the money & effort would be better spent on couple of cruise missiles (one for spare)?

ian

-always looking for the funny side ; )

Hey, its just a thought experiment anyway, right?

Dear Thread,

On re-reading the original post, the Challenge Question asks how will the dad settle the issue between the son and the daughter, not how the daughter can get better range. That's her problem.

Our problem, as the dad (whose job is clearly not to optimize his daughter's trebuchet), is to consider how the difference in gravitational attraction might settle the issue in either the son's or the daughter's favor. Am I wrong here?

Consequently, considerations of air friction and other details seem, to me at least, completely irrelevant to the resolution of this dispute, IMHO.

<flame on>

I might also take this opportunity to express my astonishment over the number of respondents are attempting to answer this question without having even a rudimentary understanding of what powers a trebuchet; especially so when 30 seconds spent googling "trebuchet" makes the principles clear enough. <flame off>

I think I need to eat some dinner to get my blood sugar up. Some of the nonsense posted here is making me overly pissy.

Sorry thread.

-e

Well said!

Of course, you know, someone will now suggest you take a vacation also, or worse, suggest an alternate destination!

Hold on. They can travel to another planet, but they are still using a trebuchet? If it were me, I would just set my phasor to "Trebuchet" and fire. Test complete, one to beam up.

Does that mean Luke, Darth, and Obi Wan can put away their light-sabres now?

-e

No,no! We'll still need the light sabers to fend off a hail of trebuchet projectiles fired by the Ewoks (friendly fire) who were aiming at the Imperial Storm Troopers chasing us!

Then again, if we got a big enough Trebuchet, it could launch our Jedi directly into their midst where the light sabers would make short work of the Imperial forces.

Funny you should mention Ewoks. When I worked at Parker Brothers (mfrs of Monopoly, et al) back in the early 80s, one of our better dark-side-of-genius game developers hung a sign over his workstation: "I love to kill baby Ewoks." And after work every day he'd go hang out at The Golden Banana along with all the other Ewok killers.

Life was good.

-e

What fun! And you quit them, WHY?

Layoffs. The game-cartridge market became saturated, and I think it was TI who bailed first and dumped their inventory on the market for $3 a game. Retailers marked-up the games to $6 - 80% less than the retail price of everyone else's product. Immediately the rest of the players followed like dominoes and dumped their product too. Then they dumped their employees.

My budget for game-design tool development dropped from $1.5 million to a little over $30k overnite. Shortly after that people were given their walking papers and only a skeleton crew remained on the Fourth Floor. I moved on to a robotics startup and eventually headed-up their AI group.

-e

Parker Brothers has an AI group? What is that, to produce an AI-based video/computer game, or to use AI to test play other games?

Or does Parker Brothers have interests ourtside of consumer game playing?

Did I say that? (Are you quite sure?)

I moved on to a robotics startup and eventually headed-up their AI group.

-e

Oh, sorry! I thought you meant a robotics startup within Parker Brothers! (picturing robots playing Monopoly)

That's very different! NEVERMIND!

No problem, Standard Template Library!

(that's what STL means to me, at any rate)

BTW, is Gateway Electronics still in St. Louis?

-e

I got bored reading all the blabber, did anyone ever mention the trehuchet is a gravity powered machine?

If you want to increase the range, you simply need to increase the mass ratio between the counterweight and projectile.

Duh.

I'm assuming it's a true trebuchet - which uses falling weights to power the arm - and not some bastardized nonsense using springs or other energy-storage mechanism.

As the focus of the puzzle is gravity, I think we can neglect the atmospheres on both planets and simply assume they're the same.

Without doing the math, I'd say the daughter is correct. Sure the projectile will fall more slowly, but it will be launched will correspondingly less vigor in a lower gravity.

Meanwhile I've gotta keep an eye on that tempermental oven. Don't wanna burn my Humble Pie.

-e

Dear Reader;

Given that same trebuchet will be used on planet Earth or any other planet (e.g. moon)with lower gravity, then the other variable is angle of trajectory to the horizon has to be same (45 deg) on both planets to see the effects of gravity, then being same angle of trajectory of 45 deg the projectile will travel farther on a planet with relatively lower gravity i.e. moon than earth.

Son is absolutely correct in this case!

Family members must also check with their mother!

Also check

Thanks

CB

Yes, but might you be assuming here that the projectile is launched with the same velocity? What powers a trebuchet? Falling weights? What do falling weights do under less gravity? Fall slower? What effects do slower-falling weights have on the catapult arm itself? It moves more slowly? What does a slower-moving catapult arm do with its payload? Throw it at the same velocity as it would under Earth's gravitational pull? Do you really think so?

Taking it to the limit, how far does a trebuchet throw a projectile in a region of Zero Gravity?

Forever? Not at all? Same as on Earth?

-e

The proposal to "just move it to another planet" is flawed, right out of the gate.

"JUST"? Maybe the boy needs to do a little research on interplanetary travel.

Generally speaking, ambient gravitational acceleration is not a parameter that can be changed by design.

Do you mean to say that the boy would actually have to be physically present on another planet to conduct the experiment??? You actually believe this is necessary???

Surely you have heard of thought experiments, yes? One of Einstein's favorite (and successful) pastimes?

(((sigh)))

-e

Being an Engineer and a dedicated Punkin Chuncker, there is only one answer.

GO COMPRESSED AIR

BUILD A BETTER MOUSE TRAP AND THE WORLD WILL KNOCK AT YOUR DOOR.

What's a Punkin Chuncker?

Probably slang or typo for "Pumpkin Chucker", which would be a person or machine dedicated to launching pumpkins along a ballistic path. Most common Pumpkin Chuckers I have heard of use compressed air expansion to replace explosive ignition of gunpowder commonly used for rapidly heating and expanding the gas/air mixture inside a large gun barrel or cannon.

The Mythbusters once used a similar device to re-create impact testing of airplane cockpit windshields by both the USAF and the RAF. One of the services used frozen turkeys fired at stationary windshield to simulate a bird strike in flight. The other service used fresh, unfrozen turkeys. There was some disagreement between the validity of the testing between the two. The myth being tested was whether there was actually any difference between using fresh, unfrozen birds, and using the frozen ones. As I recall there was no significant difference.

I believe in a later, more recent, episode they later used a similar device to launch pumpkins as well, but I do not recall for what reason.

Mythbusters revisited the frozen versus thawed chicken controversy yet again and this time firing the chickens at a target made of multiple layers of glass sandwiched together. The unfrozen chicken broke a couple of layers of glass while the frozen one smashed a hole clean through something like 9 layers of glass. The conclusion was that it dose indeed make a considerable difference with the frozen chicken being far more distructive

It is a little-known fact that the former Soviet Union's research into thawed and frozen chickens as Weapons of Mixed Destruction eventually bankrupted the country. For one thing, the thawed chickens tended to gum-up the rail guns. The only bright spot in the program was the serendipitous discovery that vodka goes well with chicken tenders.

A trebuchet uses a counterweight to propel a projectile. Therefore, in lower gravity, the counterweight will not be traveling as fast as it were on earth, and as a result the projectile will have less energy imparted to it. The range equation of freshman physics tells us that range equals ((v^2)*sin*2*(theta))/g, where v is initial velocity, theta is the angle, and g is the gravitational constant. since v is squared, it has a much larger impact on the final outcome then g. Therefore, the daughter is right.

I would say, "Settle down kids, lets see where we can buy a toy one of these, check out the internet" i.e. http://www.catapultkits.com/productinfo.php?item=11001

And then say "Let's all go down town to the big department store 'Toy Dept' then if we ask the lift attendant to press 'Down' we will have 'low' gravity and if we go 'Up' we will have high gravity"............Screams of ...."YEAAAAH, LET'S GO NOW!"

It's Christmas soon if we tip the lift attendant $5, I reckon he would not say "No" to testing one of the Store's Toys? Better still do it as a school project, and write to the manager of the Toy Department. How could they possibly refuse all that welcome custom? SORTED!

The son is correct. All objects fall at the same rate ( negating air friction eg. a feather vs a rock), so, if the projectile is fired at the same velocity, a lower gravity planet will allow the projectile to travel farther before hitting the ground.

On earth, if a .22 cal. bullet and a 30-06 cal. bullet are fired simultaneously from level guns, both bullets will hit the ground at the same time. Just like dropping them from the leaning tower of Piza. Of course, the 30-06 traveling much faster will travel farther than the .22, but gravity will pull them both to the ground. Less gravity means both would travel farther.

That's for bullets and catapults, Sister would object, and point out that a Trebuchet is just a heavy weight on a pole suspended on a fulcrum. Less gravity = Less weight = less power = less distance thrown. She would not allow you to have a bigger/ heavier weight to launch the projectile on the low gravity planet. It would upset the Space Time Continuum if anything was added or subtracted when the Trebuchet was sent through the 'Worm Hole' to the low gravity planet.

In this particular case gravity is the driving element behind the force applied to the projectile. If the gravity is reduces then you will have less force applied to overcome the same mass/momentum. Therefore lowering the gravity will cause the projectile to move slower and with slightly less momentum during flight. However, there will be slightly less gravity pulling it to the planet. Since all these factors seem to balance each other out I would have to agree with my daughter (Girls are always correct anyway even when they aren't! :-) ) The real ways to increase the firing range of a trebuchet are as follows: 1. Reduce the mass of the projectile 2. Increase the drive weight on the main arm 3. Lengthen the arm by making the trebuchet taller (thus changing lengths of weight side of main arm and projectile side of main arm) 4. Add wheels to the bottom and put it on planks so that it can roll with the change in forces thus making the forces work together. Mike

Sister; Quote:- "Yes Dad, look that is exactly what I have written in MY answer TO MY HOMEWORK! Then this stupid little Brother of mine......Typical of a MAN to come up with a TOTALLY IMPRACTICAL STUPID IDEA.....Oh yes let's send the Trebuchet to another planet.....why don't we?....have you any idea how foolish that would make me look!.......how would that stupid answer get me a good grade?.... I give up!.......MEN!"

Thanks Guest, I think you may have solved the problem for us. As Jorrie and Europium have pointed out the challenge is to settle the kids down, you write:-"4, Add wheels to the bottom and put it on planks....." If we can get both daughter and son to dissagree with dad, psychology at play, they will then aggree with each other. That will bring a bit more peace to the household at least. So Dad should suggest putting SQUARE WHEELS on the Trebuchet!

That'l fox them both, and also be another 'Toy' to play with when they find out it.s 4 real

Check out:- http://pbskids.org/dragonflytv/knowhow/knowhow_wheel.html

The shape of those planks will be critical though.

The son is definately wrong for a number of reasons. Not the least of which is the fact that the 'aim' of the problem is to increase the range of the trebuchet. Moving it several hundreds of thousands of miles away from its target here on earth is not a very simple or effective approach. From the begining of discussion it has been assumed that the trebuchet is 100% efficient. Does such a thing exist anywhere in nature? It is a useless assumption. The lever arm will not stop on a dime once its projectile is released. This continued motion is a 'waste' of PE. Also it takes energy to get the static mass of the lever arm rotating. This energy requirement is the same regardless of local gravity. Thus in a low gravity environment this requirement represents a proportionally greater piece of the now smaller energy 'pie'. The more efficient projectile flight characteristics not withstanding. Regarding atmospheric resistance. It is probably safe to assume that in general a lower gravity at the surface of the planet is likely to result in a less dense atmosphere. But Venus' case represents an exception so it shouldn't be considered a reliable indicator for our purposes.

slo

My physics goes back a long ways to when we used cannon instead of trebuchets but if I understand the way the trebuchet works, a counterweight of mass (M) is raised against gravity (g) to a height (h) thereby giving it potential energy E = Mgh. That energy is then transferred to the angular momentum of the projectile mass (m) giving it kinetic energy E = mv^2/r where v is the velocity and r is the length of the moment arm. It therefore means that v² = M/m*gh/r. But as previously stated, the range formula states range (R) = v²/g*sin(2*Θ) or R = M/m*h/r (all constants) . Therefore gravity has no effect, neglecting such niceties as friction in the trebuchet and air friction. While the frictional forces are complex, I believe that the friction within the trebuchet mechanism will just reduce the effiency and therefore pojectile velocity but velocity drops out. Drag is a function of projectile aerodynamics (ballistic coefficient) and velocity, and launch velocity increases with gravity. Since range, neglecting friction is not dependent upon gravity but drag increases with gravity, range in the presence of similar atmospheres will decrease(slightly) with increasing gravity. It should be pointed out that, at typical projectile masses and velocities drag forces will be almost negligible by comparison.

Gravity does not affect distance it only affects speed. There is a slight exception to this however. The treb arm cannot be completely ridged and higher gravity might bend it more in the cocked position. this extra spring would add to the force acting on the projectile. If you realy wanted to soup up your treb you should put it on wheels. as the counterweight swings backwards it would force the whole treb forward and add this speed to the speed of the projectile.

Mike Bush Jr

I think more information is needed now that I reread the question.We all assume that the daughter takes the posistion of the opposite of the son. She says he is wrong. He says it will farther in a lower gravity enviroment. Now is she taking the posistion that it will travel farther in a higher gravity enviroment, or travel farther on earth. <B> OR </B> does it travel the same distance. If we go on assumtion, they both are wrong. The distance traveled would be the same. Because if gravity is the only changed variable, the equation still balances out. The force of gravity on the loads is less so is the force of gravity on the projectile. Projectile will travel the same distance. Therefore the son and daughter are wrong. Now if the daughter is taking the point of there will no change in distance then she would be right.

For simplicity in the first instance we will assume that the effect of friction on the system is negligible, the projectile is released at a height h above the ground and is traveling horizontally as in the diagram below.

.

Looking at the system from the view of conservation of energy we can say M₁ falls through a distance of R₁ while M₂ is raised through a distance of R₂

Therefore the kinetic energy E₂ imparted on the projectile M₂ is

E₂ = M₁gR₁ – M₂gR₂

E₂ = g(M₁R₁ –M₂R₂) ……………..Equation 1

Calculating the velocity from this equation

E₂ = ½ M₂ V₂² …………………………….Equation 2

Therefore the velocity imparted on M2 through the trebuchet is

V₂ = Sqrt(2M₂E₂)

V₂ = Sqrt(2gM₂(M₁R₁ –M₂R₂)) …………..Equation 3

Due to the vagaries of space and time in CR4 you will need to refer to the following equations to understand the rest of this post.

Equation 4 shows what would happen to the velocity V₂ of the projectile if gravity were to change from the g_e gravity on earth to say g_m the gravity on the moon then

Now in our simplified system the projectile M₂ will fall towards the ground under the acceleration of gravity and therefore and the time t it takes to reach the ground can be calculated from Equation 5

Since we now know how long it will take the projectile to hit the ground we can now calculate the horizontal distance s it will travel in Equation 6

Now by combining Equation 6 and Equation 4 we can work out the ratio of the distance s_m the projectile would travel on the moon to the distance s_e the projectile would travel on earth in Equation 7

And since the solution to Equation 7 is always 1 it means that if we negate the effects of friction there will be no change in the distance the projectile travels under different gravitational situations.

Whilst the effect of atmosphere has been neglected here as it was in the question then it is a fair assumption that both planets would have a similar atmosphere. If this were the case then the only difference in the atmospheres would be that the one with the smaller gravity would have less density drop with altitude, but this would be negligible when you take into account the vertical distances we are talking about that could be generated with a trebuchet.

Therefore no matter how you look at it as you can see from this and other posts the only conclusion that can therefore logically be drawn is

"YOUR DAUGHTER WAS CORRECT THE REDUCED GRAVITY DOESN'T MAKE ANY DIFFERENCE."

If the trebuchet propels the projectile by a counterweight, gravity should not matter. Range would only depend on the mass of the counterweight vs that of the projectile.

But I would still take the son's side. Reason - planets (like the moon) with lower gravity will have either no air or lower air density hence lower air drag.

Grease the pivot, secure the trebuchet to the ground and use a smooth round rock. This is what any normal person would do.

Guest wrote: "Grease the pivot, secure the trebuchet to the ground and use a smooth round rock"

The challenge was how to settle the argument between your two children, not how to improve the range, but I suppose the daughter could do with some tips!

Check out:- http://www.miniaturegolfer.com/world_crazy_golf_championships_2006_results.html

It's time to make 'Crazy Golf' respectable, and invite contenders to the Royal and Ancient Links at Saint Andrews Fife Scotland. The 'Thane of Fife' has Durisdiction here (Plug). Perhaps Crazy 'Trebuchet' Golf rules could be drawn up? I noted that a Toy Trebuchet only three feet in height could hurl a golf ball the length of a football pitch. One way of 'increasing performance' is to set engineers a challenge that is of high prestige. Scotland just loves engineers, always has and always will. There is a reasonable chance this proposal if 'seconded'? ... might take off. Those kids learn from fun and games far more than from tedious homework devoid of inspiration.

Trebuchet golf! What a great idea. Unfortunately, the treb would only work well for the tee shot and maybe a second drive on a par 5 hole. I propose the following additions to the Engineering golf game:

1. Numbered woods would be replaced by multiple counterweight sizes to produce differing range for the trebuchet. Of course the player could also choose not to use the full lift of the trebuchet as well.

2. Irons would be replaced by a windup catapult which could be preset to control the angle at which the firing arm was stopped, much like the angle of the face of various irons.

3. The Putter would be replaced by a Rod Pendulum with a flat face, like a croquet mallet with a hole drilled through the handle about halfway up for a pivot pin. The pin would be fixed in a frame set on the ground and the player would merely pull the handle back until he was satisfied with the potential energy level, then release it to deliver that energy to the ball. If aime properly, a good put should result.

4. A Chipper could be made by adding an angled face and a higher mass to the Rod Pendulum.

This would make the game more interesting!

Thanks for all those suggestions, I think The Thane of Fife will consult with his lads, Charlie, George and Hugh (the youngest) The Duke of Fife's Grandaughter 'Amelia' might want an input into the discussions. As they are all Blood Royal, and it is the Royal and Ancient links, they may have the final veto. I have been busy trying to figure out if 'Square Wheels' would give an unfair advantage? The Mathematics Dept. at the local University of Saint Andrews might be able to help there? I suspect myself that with the correct size of 'Square Wheel' and the right profile of the 'planks/runners' then the entire weight of the trebuchet itself could come into effect.

The other 'Thane of Fife' is the local Constabulary, we might have to take out insurance against 'broken windows' if the kids get over enthusiatic testing their trebs?

Mmmmmm.... An initial catenary hump or two, followed by???? a hyperbolic?...no... Professor Trevor Hunt (Theology@ Saint Andrews), he's our man, we need Divine inspiration......Crazy Golf will never be the same. (P.S. Trevor will well remember the orphan crow that kids trained to pick up the ball and fly to the hole for a hole-in-one, but the rules of Golf are that if you see it, it's only a 'Birdie' (Would a Military Chaplain tell a lie?..check it out!)

"I suspect myself that with the correct size of 'Square Wheel' and the right profile of the 'planks/runners' then the entire weight of the trebuchet itself could come into effect.

SNIP

Mmmmmm.... An initial catenary hump or two, followed by???? a hyperbolic?..."

I believe that I have worked out what the track must look like for a square wheel to "roll" smoothly. At first I thought of the steps, as in the clue given, which would give a triangular cross-section (edge-view) of the track. I had even worked out the spacing and angle. Of course it had to be 90° so the square corner would fit. Then the center of the flat side had to rest on the point when the flat side was horizontal. This meant that the length of the 45° plane had to be one half of the length of the flat side of the wheel, so when it tipped over after the corner "fit" it at the bottom of the track, it would once again land the wheel at the center of the next peak, so it would fall over again, etc.

However, slippage and smoothness are concerns there. Yes it might move but it would probably go CLACKity-CLACKity-CLACKity....and shake the whole machine violently. To avoid violent slippage, and slappage, then the square wheel's corner must not only fit into a 90° groove, but immediately it should "roll" smoothly up to the peak again, and back down. Well, a semi-circle would not provide the necessary 90° tangency for the square corner, but a quarter-circle arc would do it. Then it was simple to calculate that the length of the arc must equal the length of the square side, L. Algebra yields radius, r = L÷ ∏ = .3183 x L .

Yet, this would not eliminate up and down motion, so is it "smooth"? Let's say, no. Then instead of a circular arc, the profile must be a smooth curve where the starting point has a tangency of 45°, its perimeter length must still be equal to the length of one square side, L.

Well, I started to work out the math, but saw pretty soon I would need to be looking at some serious Integral Calculus, gave up and googled square wheel , and, yes, they would be inverted catenary humps. This is a bit ironic, since I live in the city with the largest inverted catenary curve in the world, the St. Louis Gateway Arch at the Jefferson National Expansion Memorial!

STL, I spent some time in St. Louis once and went up in the Gateway Arch twice. It was amazing the first time to notice that the thing is mostly stainless-steel skin, with only a minimum of internal structure. And when they were building the Arch, they did it by building both legs at the same rate and met at the top. But when they "capped" it off, they had to wait until the temperature was just right so that the expansion of the structure made everything fit together perfectly. Quite a feat.

I also set up my camera tripod directly under the structure (as I recall there was some sort of marker embedded in the ground right under the center, which made locating the tripod very convenient.) Starting at the base of one leg, I took a series of pictures while incrementing the camera angle after each shot until I reached the base of the other leg. Then, after splicing the pictures together (too bad Photoshop and digital cameras weren't around then!), I held the resulting mosaic overhead and looked straight up. It looked as though I was under the Arch again, viewing it through a narrow overhead window. Pretty cool!

The best filet mignon I've ever had was at the Noah's Ark restaurant in St. Charles. Is that restaurant still around?

Sorry, to report, Noah's Ark, actually in nearby St. Charles, MO, shut down a few years ago. The building, and adjacent Motel is still standing, and has been used for the last few years only on the weeks before Halloween, as the "Haunted Hotel" for charity. However, this year is probably the last, as it will be demolished to build a new "mixed use" development on the site. Fittingly, the new development will also be called Noah's Ark! Let's hope someone puts in a good restaurant there!

The Arch is still standing, and probably will be until the New Madrid fault earthquake hits. I recently went up to the top with some other Alumni from UMR including a buddy who came into town for our 25 year reunion. The north end of the arch is set in bedrock, but when they excavated the foundation for the south end they found it was riddled with caves, or so the story goes. They kept pouring in millions of cubic yards of concrete, trying to fill in the caves, but it kept flowing down faster than it would harden. Finally they had to make some kind of coffer dam to hold the concrete until it would harden, then they poured the foundation on top of that, or so the story goes! Urban legend says when "the big one" hits, that foundation wil literally cave in, like a deck of cards, and the Arch will sink into that hole. No telling what the other leg would do, and whether it would then topple into the river, or onto the buildings of nearby downtown St. Louis! Could make for a good disaster story, like "Earthquake" or "The Towering Inferno". A 1968 earthquake centered in southern Illinois about 120 miles from St. Louis caused the observation deck at the top of the Arch to "bounce up and down". scaring visitors for about 40 seconds. The arch, like many tall buildings, is designed to sway back and forth somewhat with the wind, so it will bend, but not break. However, this quake was only a 5.5 on the Richter Scale. In contrast, an 1895 quake was 6.8 and estimates for the last "big one" at New Madrid in 1811 was 8.0, which caused the Mississippi River to actually flow backwards for a short time and rang church bells in Boston! Actually, the last "big one" was three 8.0 quakes and thousands of after-shocks over several months. I believe I had heard that the probability of another big one was 100% in 500-600 years, and we are nearly 200 years into that! I wonder what the probability of it hitting in the next 10 or 20 years would be?

Regarding the structure of the Arch, if you saw the film they show at the Arch about its construction, "Monument to the Dream" you should have been impressed that is was a lot more than just a Stainless Steel skin, although that skin is pretty thick and does add to the strength of the Arch, since all the plates are welded together, kind of like the exoskeleton on a bug! I believe that between the inner and outer steel walls it is filled with more steel "spines" and rein-forced concrete. I remember being impressed by how much they streched the steel "spines" to put them under tension before they poured the concrete, which makes them stronger and resists stretching that could crack the hardened concrete.

So, c'mon back! I am sure it is safe!

Thanks! Too bad about that restaurant. I forgot to mention their clam chowder was fabulous as well.

What do you think of Fallen Arches as the name of the disaster movie?