One Thing That I Think Tolkien Did Wrong and Jackson Did Right

[Galadriel55]

The Azn, I also think your drawing is inaccurate. When looking at it, the catapults higher up shoot a farther horizontal distance - which I believe is your point. But what that drawing shows is not the advantage of height, but rather that the catapults higher up have more power than those down below, which is not the topic here. All catapults, regardless of their height, throw their loads (assuming the loads are the same) the same distance, and from there it's gravity against inertia. In your drawing, your catapults far up shoot a farther horiontal distance than those down below before the inertia vs gravity thing starts (that would certainly take the load forward some more, but more downwards).

It's hard to explain this without a picture, and mine isn't accurate either, but it shows the difference between what it should be and what you drew. The diagram on the left shows three same-power catapults shooting the same loads, and on the right is what your diagram says. Because I know I won't be accurate I did not draw the trajectories all the way, but even so you can see that it's not first and foremost a question of height, but rather a question of the proportion of height against the distance in front of the catapult that is still your own castle. Certainly the overall distance the highest catapult shots is greater than the lower catapults. But the lower catapults have less Minas Tirith to shoot over too.

Secondly, the advantage in height does not mean the catapult has power to shoot farther. It does not come in with the horizontal distance the catapult projects it's load. It comes after, when the load is already losing height. In this matter I think Mumriken has the better illustration of the trajectories. They are all, regardless of height, the same parabola, but the more room they have the farther they can go - but you know that after a certain point a parabola changes vertically more than horizontally.

And lastly, I want to say again this point that is unrelated to diagrams but also important. Minas Tirith probably did not have catapults at its higher levels, especially that jut of rock in the Citadel.

[Mumriken]

the projectiles seem to shoot too randomly, which should not have happened; ancient artilleries were more consistent than what some might give them credit for. The simulation was so random that the back artillery once hit another artillery on its own side!!! Secondly, although the difference might not be much, air is less dense at 100 ft upwards. Your simulation does not appear to take variations in friction into account.

The reason they're so random is because of the friction between the rock and the catapult itself. I made lots of tests to see what the most common result was. As you can see the back catapults hardly reaches the mordor forces. Your drawing makes it look like they would reach far into the lines which isn't true. You could think of the randomness as different sized blocks of stone. Also you can't really say my simulation is more inacurrate than your drawing since it actually uses a real physic model to simulate what would actually happen. Now you could complain about the perfectly round balls or the force the catapults throws the stones away with. But that still doesn't take away from it showing that your drawing is way off.
EDIT: Also I don't think they placed catapults on that stone thing sticking out.

[TheAzn]

Hey Galadriel55 and Mumriken, most of your points would have been covered in what would have been my response to Puddleglum, Lalwende and my post afterwards.

[Mumriken]

Looking at your drawing again, they would probably reach longer if they used one of these.

But still as Galadriel's drawing and my simulation pointed out. The catapult at the back wouldn't reach the furthest into the lines. The height isn't such a big advantage except if the stones were thrown away very HORIZONTALLY and which much greater force. (Like a bullet)

But these are thrown high up in the air and therefore doesn't reach that far. (See simulation for more detail) Also the movie itself confirms the simulation and the drawing so...game over buddy.
http://www.youtube.com/watch?v=SAW2LxJRLqM

[Boromir88]

I fail to see how the movies got this one right when movies are notorious for hyping up the large battle sequences and getting physics and reality wrong (but that's also part of the appeal to movies). And your other evidence is a faulty drawing of catapult trajectories.

And on another note:

Originally Posted by TheAzn

As you have already known, 40,000 Mordorians is a very conservative number; heck even 80,000 is still considered to be a moderately conservative number. With this in mind, the Mordorians/Morgulians/Haradrims/etc. would have to be far closer to Minas Tirith than 7 miles.

Well, 40,000 is the the low ball estimate, but 80,000 is at the very high end. The estimate for the Pelennor Fields is really 40-65 thousand. Sauron's force coming out of the Black Gate was 70,000 and I haven't seen a case to show his combined Pelennor Fields force would be greater. The movies 250,000 or whatever it was was again, the film business liking the large, epic battles.

[Galadriel55]

There is also this point which was mentioned before but not really elaborated on: the Mordor forces were firing heads over the walls of the first couple levels, and some small loads that would explode into fire. Gondorians were most likely firing large blocks such as you see in the movie. A catapult shoots lighter loads much farther than heavier loads. Gondor did not, for a fact, have anything that would equal to Mordor's explosive loads, and what's the point of firing heads at the orcs?

As for the preparation Denethor would have done, remember that Minas Tirith is unassailable. You can't break the walls with catapults. The Gondorians laughed at Sauron's catapults, because they would not do much harm to the walls. Denethor does not need to prepare a defence for a catapult attack, he has one already.

ETA: and yes, as Mumriken said, height could give advantage to, for example, archers, who shoot on a much flatter parabola. But not so much catapults.

[TheAzn]

Originally Posted by Mumriken

Looking at your drawing again, they would probably reach longer if they used one of these.

But still as Galadriel's drawing and my simulation pointed out. The catapult at the back wouldn't reach the furthest into the lines. The height isn't such a big advantage except if the stones were thrown away very HORIZONTALLY and which much greater force. (Like a bullet)

But these are thrown high up in the air and therefore doesn't reach that far. (See simulation for more detail) Also the movie itself confirms the simulation and the drawing so...game over buddy.
http://www.youtube.com/watch?v=SAW2LxJRLqM

Although I did admit that Mr. Peter Jackson made more sense than Professor Tolkien in this particular scenario, that does not mean that I agree with Mr. Jackson with everything, even in the scene where I agree with him the most. So no, it is not game over yet, good sir.

[TheAzn]

Hey Boromir88, Galadriel55, and Mumriken, I would like to kindly make a request on what I have already implied.

Can you guys please wait until I have adequately respond to those before you before you guys post anything else? Most of your points are pretty much the same, and I my explanations to the other posters could have refuted your arguments beforehand. I think that it would be best to makes my arguments very clear and honest by responding to each and every one of you. However, when you guys keep on adding new posts when I am not even finished, this "duty" becomes quite hard to do.

However, I think that it is also wrong to make you guys wait and I certainly do not have the right to tell you guys to do anything. Since I will be gone for half a day, I will make my revised arguments right now on this very post. I think this post will refute most of the arguments, and you guys can respond to it. I might expand on the arguments later on.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Gondor Can Use Small Projectiles Too!

Denethor and the Gondorians do not need large and heavy projectiles to destroy the engines of the Mordorians.

A stone was hurled (by hand?) from Barad-dur, and it struck an interesting person. That person was no other than Anarion. Being a of royalty he most likely had a very strong armor, and on his head he wore the "Crown" of Minas Arnor, which is most likely a very strong helmet. The (hand size?) stone broke through the helmet and killed the tough man Anarion instantly. Of course, Barad-dur is much taller than Minas Tirith, being at 5000 feet at the top. However, the stone need not be cast down at that height. Still, I will admit that the case is vague.

But thankfully for me, there is a very clear case, and that would be at a place called Isengard. At the Orthanc, Grima Wormtongue became so desperate that he threw the Palantiri at the people down below. Grima Wormtongue is not retarded; though Palantiri might miss, if the stone were to hit anybody that person would die instantly, even the tough Gandalf. You can test this yourself if you don't believe me ( I do not recommend it). The height of the Orthanc is 500 ft AT MOST. Grima Wormtongue probably threw the Palantiri at a much lower level, probably at 100 to 300 ft.

What is interesting is that, strong or weak from demoralization, Grima uses merely his human strength to throw the stone. This means that the stone was not very heavy. Yet, Professor Tolkien rightly implied that such stone, from height, can cause huge damages.
With this in mind, let's talk about Minas Tirith. Like the Mordorians, the Gondorians can also use light projectiles to gain range. At the end of the pathway, the projectile increases in lethality thanks to the acceleration caused by gravity. To clarify: a projectile fired from the bottom wall has about 100 ft of vertical acceleration; a projectile fired from the Second Wall has 200 ft of vertical acceleration; and a projectile fired near the frontline from the "bow of the ship" has about 700 feet of vertical accleration!!!!

As people know, a mere crack in a cannon can make it useless, and requires a relatively long time to fix. If the Mordorian artilleries have just any of their essential parts destroyed, then those artilleries cannot be fired for a while. This is a mighty advantage for the Gondorians.

It is true that Tolkien did not explicitly mention any aritilleries on the jutted rock. However, having artillieries there is the only option that made sense, especially when we are talking about artillieries that are just larger than 2 people.

Examples of portable artilleries built by normal people:
http://www.youtube.com/watch?v=GT1Q09BsMBo
http://www.youtube.com/watch?v=hsY7a...eature=related

[Galadriel55]

Once the Mordor artilleries have fired their explosive loads, quickly followed by the human heads, they've done their job and it doesn't matter who or what hits them. They can go back to Mordor for all the difference it makes.

And once again, because you still refuse to listen even after all those times that all those people told you so, Gondor's catapults could not reach the Mordorian ones!!! Could not. That's that. You say yourself that Gondorians aren't stupid, that they too can use lighter loads, but the situation for whatever reason was such that whatever they did did not help them in time. And that's that. Period. What's your point here?

As for the palantir, it can kill one person when falling from a height. Big deal, when you have many thousand orcs running at you.

[TheAzn]

Originally Posted by Mumriken

The reason they're so random is because of the friction between the rock and the catapult itself. I made lots of tests to see what the most common result was. As you can see the back catapults hardly reaches the mordor forces. Your drawing makes it look like they would reach far into the lines which isn't true. You could think of the randomness as different sized blocks of stone. Also you can't really say my simulation is more inacurrate than your drawing since it actually uses a real physic model to simulate what would actually happen. Now you could complain about the perfectly round balls or the force the catapults throws the stones away with. But that still doesn't take away from it showing that your drawing is way off.
EDIT: Also I don't think they placed catapults on that stone thing sticking out.

Since I have already typed them out, I will repeat what I have said to miss Galadriel55.

Originally Posted by TheAzn

Hey Galadriel55. Besides a few things, I pretty much agree with what you said, but there are a few problems. Before expressing your opinion on who is more accurate, please have the courtesy to wait and ask for more clarity. Before I continue, I would like to say that I am at fault also. I should have been more clearer in my OP. As stated in my later post, there is less air friction the higher up you go. My drawing takes this fact into account. But more important is the fact that the catapults in my drawing were meant to not shoot the same loads. I guess I have to make some edits for my drawings.
And yes, I am saying that the Gondorians can use even lighter loads than the Mordorians to bombard , while the Mordorians can only cause “little” damage with their light but still heavier loads. I will explain more below in what will be my great responses to the general arguments made against me.

Originally Posted by Mumriken

Now you could complain about the perfectly round balls or the force the catapults throws the stones away with

Well, what I am going to do is not really complaining, more like pointing out your mistakes. If you were to think things through a bit more carefully, you would see that the Gondorians would, if Tolkien is consistent, have very likely produced exactly those round stones or any other aerodynamically efficient projectiles - especially if those projectiles are small and thus do not require much resources.

As stated already, just because I agree with Peter Jackson does not mean that I agree with him entirely. I agree with Peter Jackson more than Professor Tolkien on this particular event because it makes sense for Gondor to be able to reach the Mordorian artilleries. I have never claimed that Peter Jackson was right by making the Gondorians shoot boulders against the Mordorians. My original drawing was supposed to portray the flight path of aerodynamically efficient projectiles.

[TheAzn]

My Grand Refutation to the General Counter-Arguments Made Against Me
There are four main points that I would like to make that would refute most of the people’s arguments on this thread. These four points are, in no particular order: (1) Despite having strong walls, the Gondorians still have many overwhelming incentives to use the best artilleries; (2) Gondor can use light projectiles too, and can amazingly still create more damages than the Mordorians; (3) If Tolkien is to remain consistent, then there is nothing in Minas Tirith that would prevent the use and deployment of artilleries; (4) There are limits as to how bad the Gondorian artilleries can be and there are limits as to how good the Mordorian artilleries can be.
Due to the nature of the debate, I will provide multiple pictures. Because there are restrictions on how many pictures I can put down per post, my arguments will come in multiple posts.


Let us now begin. As you can see, the simplified picture below shows a trebuchet-like artillery firing a piece of projectile. The dotted lines represent the forward path of the projectile if it was to be fired from a ground level position. When the projectile crosses the dotted lines, that would have been its maximum range at ground level.


With the basic things - I hope - starting out clearly, let me show you a revised edition of my first drawing. In this revision, you can now clearly see that the parabola is not symmetrical, and the last half of the trajectory tend to drop down faster than the rise of the first half. The walls will not be exactly equal in height, since Professor Tolkien merely stated that each walls were “about 100 feet higher” than the other. This, in case you are wondering, is why the 1st wall is about 70 feet in height in my picture.


As can be seen from the dotted lines, each artilleries on the Gondorian sides are firing the same load. The most interesting question, then, is why are the Gondorians are outranging the Mordorians when the Gondorians have weaker artilleries. There are 2 answers and both should have been easily understood from the beginning. The first answer is that the Gondorians have a sheer advantage in height. Past the dotted lines, the projectiles fired from Minas Tirith was able to gain more range, even when they are going down faster than going forward. This is an advantage that the Mordorians do not have.
The second and more important answer is that the Gondorians are using lighter projectiles, even lighter than what the Mordorians are using. We know that a human head weigh around 8 to 12 lbs. So, assuming that the Mordorians did not fire multiple heads per round, 8 to 12 lbs are the smallest weight of projectile possible for certain Mordorian artilleries that are firing. The firebombs are a bit tricky, since Tolkien can simply invoke magic. However, just in case some people are curious, a mere gallon of “average” oil weigh from 6 to 8 lbs.
The Gondorians can simply respond to this threat by shooting stones weighing from 5 lbs to 4 lbs to 2 lbs. In other words, these stones would weigh- respectively speaking - twice, thrice and six times less than the heads thrown from the Mordorian lines. The Gondorians should have no problem reaching the baddies.

But this begs a question. How can a mere 5 lb - much less a 2lb - stone do great damages? The answer to the interesting question is very clear: please look at the blue lines on the above drawing. The Gondorians have extra help from friendly gravity, something that the Mordorians do not. Of course, lines might be confusing to some people here, so I decided to call for help. The people that I just called are Saruman the White and Mr. Grima, who are now good guys. They will teleport here to the Pelennor Fields, and the Orthanc will come with them. Hi Ho!!!!!! They are here now, in the middle of the Mordorian forces, throwing 5 lb stones from the various heights of the Orthanc.

The picture below shows the Gondorian artilleries from the bottom 3 levels firing 5 lb stones, while Grima, at the same heights, redeems himself by throwing 5 lb stones at another piece of Modorian artilleries.

[TheAzn]

My Grand Refutation to the General Counter-Arguments Made Against Me(Cont'd)

The picture below shows the Gondorian artilleries from the upper 3 levels firing 5 lb stones, while Grima, at the same heights, redeems himself by throwing 5 lb stones at another piece of Modorian artilleries.


Can you imagine the sheer speed gained from acceleration at various heights? If you are in awe, then you have reached the right conclusion. A 5 lb stone dropped from 500 feet will gain a speed close to its speed in vacuum - 101 miles per hour.

What can an object traveling at 101 miles per hour do? Below is a video showing what a mere 15 lb wiggly wood, traveling between 70 miles per hour to greater than 100 miles per hour, can do.
http://www.youtube.com/watch?v=_8hs1...6157DAE738106D

The holes might look small from afar, but it is actually sizeable. The same force from the stones would have likely created gaping cracks in all of the Mordorian (mostly wooden ) artilleries, rendering them useless.

Of course, there is a limit as to how light the projectiles could go. For the lighter the projectiles the more the air resistance can influence it. However, the shape of an object is more important than weight when it comes to air resistance. A 100 lb Flatboard will not fly as far as a 2 lb Round Stone when fired from the same “source”.
I am also aware that an object in real life cannot accelerate by gravity forever. A dropped object , as you know, accelerate due to the weight of the said object. The lighter the object, the faster air resistance can stop acceleration from gravity. The result is the final velocity called the terminal velocity, and this velocity will not change until the object hit the ground ( and it will change to 0 of course). However, I am willing to say that even a 2 lb stone would still have an amazing terminal velocity when dropped from 400 feet and above.

The Siege of Gondor should have been a bad day for the poor Mordorian siege equipments, with no additional help from gravity. Those heads, with their surrounding soft flesh, can serve not much purpose besides making people sick and tiring their shield arms. Meanwhile, things were very favorable for the Gondorians. Under no circumstances should the Gondorians have any problems using light projectiles to reach the trenches, and then let friendly gravity enhance greatly the destructiveness of the projectiles.

[TheAzn]

My Grand Refutation to the General Counter-Arguments Made Against Me(Cont'd)

Sadly, there are still some counterarguments for my refutations. Since the artilleries cannot destroy the lower Great Wall of Minas Tirith, why would the Gondorians waste time and resources to take care of any artilleries of their own? The correct answer, like always, should have been easily understood to readers from the beginning. Without artilleries, there are still things that are very dangerous in Middle Earth; anything can happen and having extra range, especially in a life or death struggle, is exceptionally necessary. False pride cannot explain Gondorian behaviors, unless false pride means “sub-human intelligence”.

Gondor has been aware of the Haradrims and their Mumakils for an exceptionally long time before the War of Ring. And it was known to most people that the Mumakils were roaming openly in Ithilien, threatening Gondor. Indeed, the Mumakils were one of the animals leading the assault on Minas Tirith. Description for the height of such creatures varies. However, when adding their height and that of the towers on their back, the Gondorian Archer’s advantage in height disappears to nothing. The Haradrims can afford huge casualties; the Gondorians cannot. Thousands of arrows later, and the Oliphaunts might feel a bit irritated. Meanwhile, the Haradrims rained death upon the Gondorians, all before the orcs and trolls even reach the walls. Only Fools do not want artilleries in such a scenario. With artilleries, cheap stones can crack the skull, break the spine, or knock out the pillars supporting the towers of the Mumakil. Sure, I might be overestimating here, and the stone might cause nothing but irritation. But at least the beasts were irritated before the Haradrim can even rain death upon the poor Gondorian archers. The archers of Gondor can then focus on what archery is most efficient doing: killing masses of orcs.

But let’s continue assuming that the Gondorians like to waste arrows. As implied, there is another tough type of creatures, and it would be the trolls. After the Mumakils were finally irritated enough to go insane and trample on some forces of Mordor, the trolls rushed forward. With abundant arrows they are already tough enough to kill,and the poor Gondorians have expended most of their arrows. Although the lower walls cannot be broken, it can be climbed over. Needless to say trolls in close quarter combat, a type of warfare in cities, are very dangerous. If killed, they can still win by collapsing on you. These large creatures could have been shot down beyond arrow range and before they could reach the wall. The incentives for High -Tech Artilleries grow exceptionally strong. To reiterate, only Fools do not want artilleries in such a scenario.

The final and greatest point on the subject of incentives would be this: round stones are actually cheap to produce and do not require great resources. Please watch the video below. You may skip to 1:34 minutes and watch up to 7:23 minutes.
http://www.youtube.com/watch?v=jn2wbZow8cw
As you will see, it requires only one person to carve out a 57 lbs stone in just a few days. Amazingly, a 57 lbs (26 kgs) stone might be even cheaper to produce than a single arrow!!! And no, the artilleries themselves need not be more expensive than bows. Although requiring more materials, certain artilleries can be built within a year, while certain bows needed 2 to 3 years to dry out. The incentives for the Gondorians to prioritize strong artilleries before any other weapons is now indescribable. Only the most foolish of Fools do not want artilleries in such a scenario.
.................................................. .................................................. .

I have pretty much refuted the arguments that, because of their strong walls, the Gondorians have no incentives to build artilleries. When the Gondorians are so stupendously outnumbered, they have every incentive to increase range.


But what about the problem of space? I have to say that this is not a problem for my arguments at all. Most of the population has been evacuated or simply uninhabited from earlier periods. Obviously prudent steps should have been taken by tearing down non-essential buildings. I am not talking only about removing fire threats from the engines of Mordor. There are also accidental fire threats that could happen from the inside as well. Also, if the orcs managed to climb or break in, you do not want these evil beings to be sheltered from the arrows of the above levels. Certainly, some buildings need to be spared, but I believe that a lot of space could be created. Of course, Minas Tirith might not look pretty once Gondor wins the war, but Gondor would have to win first. If artilleries cannot be placed on the walls, then they can be placed on the newly created “ground” of each level. Since the “ground” rises higher at every level, the effect is still the same as the artilleries being placed on top of the walls.

One question remains about the problem of space. Can artilleries be placed on the top of the Ship Rock? Tolkien did not explicitly state whether or not artilleries were posted there. However, the sheer amount of evidence support my position that they were indeed posted there. Before continuing on, let us look at the quote below.

Minas Tirith
For the fashion of Minas Tirith was such that it was built on seven levels, each delved into the hill, and about each was set a wall, and in each wall was a gate. But the gates were not set in a line: the Great Gate in the City Wall was at the east point of the circuit, but the next faced half south, and the third half north, and so to and fro upwards; so that the paved way that climbed towards the Citadel turned first this way and then that across the face of the hill. And each time that it passed the line of the Great Gate it went through an arched tunnel, piercing a vast pier of rock whose huge out thrust bulk divided in two all the circles of the City save the first. For partly in the primeval shaping of the hill, partly by the mighty craft and labour of old, there stood up from the rear of the wide court behind the Gate a towering bastion of stone, its edge sharp as a ship-keel facing east. Up it rose, even to the level of the topmost circle, and there was crowned by a battlement; so that those in the Citadel might, like mariners in a mountainous ship, look from its peak sheer down upon the Gate seven hundred feet below.

Other than the problem of space, there is also a concern about whether or not the rock can handle the forces created by the artillery pieces. Thankfully for me, sheer evidence supported the opinion that it most likely can handle the ordeal. The rock was already securely there, and there was also work on it by a mighty craft. These special craftsmen were most likely the same ones that built the mighty Lower Wall.

But even without the explanation of “magical craftsmen”, there are plenty of places in the world today where heavy items are safely placed on what seems like precarious lands. One such example would be Christ the Redeemer, which was built while Tolkien was alive.

[TheAzn]

My Grand Refutation to the General Counter-Arguments Made Against Me(Cont'd)

So there are no problems about the rock crumbling, and now we are back to talking about the issue of space. Thankfully for me again, it is also clear that space is not lacking on the Upper Deck of The Ship. The blatantly obvious clue here is the word “mariners”; even at the very tip of The Ship, there is enough room for multiple persons. Since the rock is described only as the bow of the ship and not as an entire ship (bow and stern), it is obvious that the rock would only widen when you move back towards the Tower of Ecthelion; there can only be more spaces behind the “front tip” of The Ship. Many artilleries can be placed just a bit behind the Second Wall, and they will be 700 feet up in the air.


Indeed, all of the precise replicas of Minas Tirith that I have ever seen shows that many people can be fitted on the Upper Deck. All the things that might hinder the placement of artilleries on The Ship has now been eliminated. If the Gondorians still won’t place the artilleries on such a great height, then they are far below “sub-human” in terms of intelligence. The sheer amount of advantages that they are throwing away is appalling.

[TheAzn]

Originally Posted by Galadriel55

The Azn, I also think your drawing is inaccurate. When looking at it, the catapults higher up shoot a farther horizontal distance - which I believe is your point. But what that drawing shows is not the advantage of height, but rather that the catapults higher up have more power than those down below, which is not the topic here. All catapults, regardless of their height, throw their loads (assuming the loads are the same) the same distance, and from there it's gravity against inertia. In your drawing, your catapults far up shoot a farther horiontal distance than those down below before the inertia vs gravity thing starts (that would certainly take the load forward some more, but more downwards).

Hey Galadriel55. Besides a few things, I pretty much agree with what you said, but there are a few problems. Before expressing your opinion on who is more accurate, please have the courtesy to wait and ask for more clarity. Before I continue, I would like to say that I am at fault also. I should have been more clearer in my OP. As stated in my later post, there is less air friction the higher up you go. My drawing takes this fact into account. But more important is the fact that the catapults in my drawing were meant to not shoot the same loads. I guess I have to make some edits for my drawings.
And yes, I am saying that the Gondorians can use even lighter loads than the Mordorians to bombard , while the Mordorians can only cause “little” damage with their light but still heavier loads. I will explain more below in what will be my great responses to the general arguments made against me.

It's hard to explain this without a picture, and mine isn't accurate either, but it shows the difference between what it should be and what you drew. The diagram on the left shows three same-power catapults shooting the same loads, and on the right is what your diagram says. Because I know I won't be accurate I did not draw the trajectories all the way, but even so you can see that it's not first and foremost a question of height, but rather a question of the proportion of height against the distance in front of the catapult that is still your own castle. Certainly the overall distance the highest catapult shots is greater than the lower catapults. But the lower catapults have less Minas Tirith to shoot over too.

If we are talking about the same load, then yes you are quite right. But just to be perfectly clear on one thing, even the catapults placed on the lower wall will have the advantage of height, if Tolkien is consistent with his writing. The advantage will not be humongous, but it will be significant if the height is somewhere around 100 ft.

Secondly, the advantage in height does not mean the catapult has power to shoot farther. It does not come in with the horizontal distance the catapult projects it's load. It comes after, when the load is already losing height. In this matter I think Mumriken has the better illustration of the trajectories. They are all, regardless of height, the same parabola, but the more room they have the farther they can go - but you know that after a certain point a parabola changes vertically more than horizontally.

Besides your opinion on the accuracy of Mumriken’s illustration compared to mine, I pretty much agree with you. Yes, after a certain point, the parabola changes vertically more than horizontally.

And lastly, I want to say again this point that is unrelated to diagrams but also important. Minas Tirith probably did not have catapults at its higher levels, especially that jut of rock in the Citadel.

This is the greatest problem with your post. The evidence is actually quite to the contrary; catapults, if Professor Tolkien is to remain consistent, would have very likely been placed at the higher levels, and the jut of rock is no exception.