# Assaulting a floating city: Two-Stage Ballista solution?

An evil empire is besieging a floating city. While not particularly high off the ground, none of the siege weapons can reach it effectively. They also need a way to get troops into the city.

An imperial engineer comes up with a design:

A large ballista fires a glider as far as it can into the air. The glider is essentially a smaller, winged ballista that fires once it reaches its peak height. The idea is to have the second projectile tied to a rope, allowing soldiers to climb up to the city.

Some background info:

• The world is low magic fantasy. The city floats by unknown powers, and there are no magical flying machines.
• The city is situated on a plain, so the ground around it is relatively flat in every direction.
• The tech level is between medieval and industrial revolution, however gunpowder and airships do not exist yet.
• The height at which the city floats at is not set in stone. This can be adjusted.

EDIT:

• The city is stationary.
• Each nation is a single city state, barring the Empire, which is consuming nations one by one.
• The nation being besieged is a nation of scholars, scientists and strategists - some of the greatest minds in the world.
• In order to minimize reliance on the ground, the city has a self contained water system.

The inhabitants have a large view radius, and thus have fair warning of the approaching army. All non military personnel are evacuated and take refuge in friendly neighboring states, leaving only the small but efficient army to man the battlements. The king and his son remain to command the troops. With only a few hundred men, the preserved food stores would allow the defenders to hold out long enough to make a prolonged siege very costly.

At the end of the day the city must fall for the story to progress, but I can't think of a way without airships/magic to do this. I could allow gunpowder and have them blast it out of the air, but I need enemy soldiers to infiltrate the city for plot reasons.

Is this two-stage ballista solution plausible, and if not, what alternatives/alterations should be looked at?

EDIT: I'm strictly looking for mechanical solutions. Assume that the defenders are extremely loyal and have the greatest disdain for the enemy

Notice: Apologies for not replying to comments etc. I kinda have a medical emergency, but will go through everything once it's passed. Thanks for all the answers!

• Does it have to be an engineering solution? If not, the promise of a heap of gold and/or power will surely tempt some city-dwellers to lower a rope at night. – Cyrus Mar 10 '17 at 11:19
• A fancy two stage balista would no doubt look good in Hollywood CGI, or in a rendered by hands of a skilled artist. In written form though, it sounds... a bit contrived. Help from an insider is so much easier. – Guran Mar 10 '17 at 11:35
• @Cyrus Yes I suppose I should have stated that in the question. Though editing now invalidates an answer. The civilians are evacuated beforehand, leaving only the loyal troops. – Lu22 Mar 10 '17 at 11:39
• I don't know about multi-stage ballista... But how about multi-stage kites, raised by drawing them along the plain by cavalry units. The topmost kite could be equipped with a hook and latch onto the city as soon as it reaches it. Then the rope to climb is already there! – M.Herzkamp Mar 10 '17 at 15:35
• In the Siege of Masada the Romans got to the entrance of the fortress ---located on top of a cliff--- by building a 114 meter high assault ramp leading up to the cliff. It took them 2-3 months. – epa095 Mar 10 '17 at 17:14

## 15 Answers

A glider light enough to be launched a good distance in the air, yet rigid and resilient enough to bear its own siege weapon, seems a bit infeasible. I suggest altering the plan in a couple key areas.

For one, using a trebuchet or similar design to launch the initial payload would likely be more practical. These are by far the best distance launchers available to the low-fantasy medieval theme.

Instead of the glider, let the payload tumble freely, but have an exceedingly clever mechanism built into it that will disengage once its airspeed falls low enough, at the apogee of the launch. Said mechanism will release one or more parachutes to unfurl and slow the secondary stage further. This accomplishes two goals. The first, the drag from the parachute as it start to fall will help assure the dangerous end of your payload is pointed upward. And secondly, the tension from the connecting lines will pull on the attachment points to trigger the secondary weapon to fire upward into your target, or over the wall, depending on whether you trust your low-tech rocket scientists to get the orientation close enough to launch at an angle.

The descending second stage will carry the cable down to the ground, possibly on a spool to allow it to uncoil cleanly. As a bonus, if your engineers are up to it, it could double as a crude elevator with man-power ratchet mechanisms to lift your forces up to the target!

• Great answer, and great name. Welcome to Worldbuilding. – kingledion Mar 10 '17 at 15:54
• I may get flak for picking a +2 answer, but this keeps with the over the top approach I was going for, while still having some practicality. Looking forward to your other contributions. – Lu22 Mar 23 '17 at 13:33
• @Lu22 I am surprised but pleased that this idea fit the best, especially after the evil-genius potential of Cyrus' answer. – thepizzaelemental Mar 23 '17 at 14:51
• @thepizzaelemental Yeah there were a bunch of brilliant answers, but unfortunately I can only choose one and upvote the rest. – Lu22 Mar 24 '17 at 10:57
• Step 1: start with the superior siege machine Step 2: ???? Step 3: profit – AAlig Sep 17 '18 at 19:36

The two-stage ballista with a glider sounds very cool but is bound to fail, mostly because a primitive glider would either be too fragile to survive the launch or too heavy to get anywhere near as high as a simple harpoon fired from the same ballista. Even modern gliders can only gain altitude when they fly in rising air (thermals or wind over hills/mountains) to use, so it's highly doubtful your first generation experimental contraptions could.

Your goal is to have enemy soldiers infiltrate the city without too long a wait, so simply building a hill next to the city for your siege engines is not an option.

Still want to fly? Try a balloon

If you are okay with gliders, you might want to consider a primitive hot air balloon, the type they could fill from a bonfire underneath the city and let it ascend while still attached with long ropes. You'd need one or two suicidal fools to get in and fire grappling hooks from crossbows and hook them to the city's underside somehow.

Plan B: Ballista with firework rockets!

In China, fireworks existed for centuries and were even used in warfare to a limited extent. It wasn't until much later that people figured how to really make gunpowder explode with much force.

If you are willing to allow "low-explosive" powder that doesn't work for cannons or guns, you could have rockets to carry up a rope or wire to attach to the city. And if the rockets by themselves are not strong enough to cover the distance, you could first launch them part of the way with your ballista.

The nice part here is that you can make this very rare and exclusive. Perhaps the evil emperor traded a fortune away for a handful of these from the only nation that knows how to make them or maybe the Alchemist that created them dies in a testing mishap. Either way, the sky is not going fill with rockets soon, it's a very special one-time affair, saved for assaulting the floating city.

One addendum: The smokescreen

Both these solutions are not-quite-stealthy, especially if the defenders keep watch from the bottom of the city. To protect them from being spotted and generally make life unpleasant for the defenders, the evil army should bring wagons with large kettles of oil that they burn to make a smokescreen, moving the wagons as needed to be upwind of the city. They can substitute grass/leaves etc if oil is in short supply.

• Plan B plus the smokescreen are a pretty creative solution. I have rather rare reagents available that could theoretically fuel rockets over short distances. The scene I imagine is quite hellish and would cripple the defenders morale. – Lu22 Mar 10 '17 at 12:55
• If you want hellish, the wagons could be burning the bodies of city inhabitants that were too slow escaping. I focused on the engineering, not the evil part, but there are options here... – Cyrus Mar 10 '17 at 13:19
• :| Brutal. I love it – Lu22 Mar 10 '17 at 13:29
• It is a low-magic fantasy world. Creating fire, or even just heat, seems within the realm of possibility for a world where a city mysteriously floats in the sky. The empire doesn't even need to have their own magicians. A single captured family containing a magician should be able to weave together whatever spell is necessary to generate lift for some gliders, and probably even protect the glider during launch. I mention family because of economics 101: people only do things if their incentives are sufficient. – Chris Pfohl Mar 10 '17 at 14:44
• Indeed, there's no way in heck that Ballista1 + Ballista2 would be able to shoot a bolt higher than just ballista 1 could shoot a bolt. The only reason rockets to space are multi-stage is because they have to carry their fuel with them, and there's no point in continuing to carry a big empty tank after you've expended all the fuel in it. And rockets are almost entirely fuel on the launch pad. – Shufflepants Mar 10 '17 at 15:35

Start a fire under the city. Spend a week or two cutting down surrounding wooded areas and breaking up nearby buildings. Pile all the wood up under the city to create a giant bonfire. Wait for a day with relatively light wind and set the whole thing ablaze. If the city was low enough to the ground it would turn into a giant frying pan, but if it were higher the main danger to the garrison would be the smoke, completely filling the city. Fires create updrafts of the sort depicted below:

While the garrison closes every window and retreats to the deepest recesses of the city to escape the smoke your troops can use gliders to ride these updrafts straight to the city. They would perhaps need some magical or technical protection against the smoke, but if they stay near the edges of the column they could likely manage it.

# Tackling an enemy's strength head-on is a good way to lose

The city's great strength is it's inaccessibility. That's difficult and expensive to overcome. Don't bother.

The city's great weakness is it's population...who are conveniently on the ground in nearby states. Attack that relatively unprotected population. A truly great, scene-chewing evil emperor should revel in such malevolence, and in the many opportunities for deceit and horror and betrayal implicit in that strategy. Why would any evil empire worthy of the name hesitate to raid nearby neutrals?

The city's King has chosen an unwise strategy - his lone battalion of soldiers do not seem to have enough mass for an effective sortie, and can be bottled up by a mere brigade of the empire. The scenario seems to require that the City has minimal useful advantage from it's elevation - which is a shame, as simply dropping hot/hard/sharp/poisoned/infected/gruesome objects onto the vast army gathered below seems an obvious defense.

The evil emperor seems free to send the vast bulk of his forces raiding. In a year or two, the siege will be over, and the empire will own the floating City of The Starved (or the City of The King Who Caved) while the city's children toil in misery below as the emperor's slaves.

I know this isn't the answer you're looking for - it's not a way to storm the city, there's no fantastic battle, no heroic defense. It's merely the cheapest, easiest, and most foul way for evil to really grind it's victory into the reader's face.

• Assuming the city can't move, that is. It would be hard to besiege something you can't effectively surround. – Andrew Mar 10 '17 at 20:28
• if the city can move sideways then there is no way to take it, it can just fly to another continent. – John Mar 10 '17 at 20:38
• I like this answer, the city as it sits is more an annoyance then a threat, and it would be even more vulnerable to siege than normal castles, and the stuff an empire would really want, the land and people, is undefended. – John Mar 10 '17 at 20:42

The way you describe it, the city is floating, the height at which it does can be adjusted, but it doesn't seem like its position can. Like regular ground-based cities, yours will have some problems regarding food and water supply, possibly greater than what a ground-based city might have.

How does the city supply itself? Eventually the people inside will have to get to the ground to get food and water; even if the farmland necessary for the city to sustain itself was floating, it would still need water, and its inhabitants would need it too.

The best and safest way to capture such a city would be to surround it, capture its water/food supply and just wait. Humans tend to require water, else bad stuff happens.

If you need your soldiers to infiltrate the city, you could set up a desperate run for food/water by the people inside, have them captured, and then send back up your men.

• Yes that is the basis of a siege. I want the city to fall quickly though, and have measures against starving out. Though adding the elements that negate a prolonged siege to my question would invalidate your answer - something I'm not happy with doing. With your permission I will add more detail to the question. – Lu22 Mar 10 '17 at 11:44
• by all means, a more detailed question gproduces more useful answers ;) – Alex Zuan Mar 10 '17 at 11:49
• Done. Hopefully I covered everything this time. – Lu22 Mar 10 '17 at 12:09
• If it's low enough it should still rain on the city. Not to mention they could collect water vapour from clouds. But capturing the food would work if you also ruined the crops somehow. – Pharap Mar 10 '17 at 16:23
• even if they have food the city will not have fuel, and most importantly no way to resupply anything. so just take over the ground under them and move in settlers, the more your city tries to defend the ground below the more desperate they become as their supplies get used up. Plus every time the descend to drop off troops they bring themselves inside normal ballista range. A siege would not last long unless the city refused to fight, in which cases who cares about them. – John Mar 10 '17 at 20:15

I think your best option lies with an elongated trebuchet. A normal trebuchet would look like this:

o-------------[]
|
|


It can shoot heavy stones to quite large distances, however, if you substitute the part left of the fulcrum by a much thinner, longer arm:

··················----[]
|
|


You can esentially launch a much lighter projectile (such as a small harpoon tied to a thin rope) at several times the speed, achieving distances nearly an order of magnitude higher.

• A longer limb prevents the counterweight being lifted as high, and reduces the energy that is imparted into the projectile. A trebuchet does have a sling at the end of the limb that extends this arm anyway... If you raise the fulcrum to allow the limb to be pulled down, you are essentially saying "build a bigger trebuchet". – Whinja Mar 10 '17 at 17:13
• You don't have to raise the fulcrum. The counterweight is designed to rest on the floor just in the middle of the fulcrum, no matter how long is the other arm. The problem, of course, is a very long arm will weight a lot or, if you make it thinner and lighter it won't resist the stress. But for any trebuchet able to launch a X weight at Y distance you can adapt the very same trebuchet to launch a X/10 at 6 or 7 times Y distance (it doesn't achieves 10 times because of many mechanical problems, but still). – Rekesoft Mar 13 '17 at 8:36
• virtualtrebuchet.com/#simulator - Supposedly, a trebuchet with a 60 foot long arm and 10 tons of weight can put a 500 pound projectile 600 feet in the air using a release angle of 70 degrees. Weight-wise, that could be a 200 pound rock with 300 feet of 2" manila rope attached to it. We're gonna need a bigger boat. – Mazura Mar 14 '17 at 5:43

First, this city is impossible to assault. If a city is immune to siege engines due to being able to go higher, there is absolutely no way your army can climb up even if somebody can get some ropes up.

Historically, the solution to assaulting the unassailable has been to make it assailable by building a pathway that can be used for assault. Masada and Tyre being the easy examples. Persian conquest of Babylon by diverting the river is also nice.

Your city seems immune to even that. But then again... The analogy would be to not assault it when its floating, but to first get it to come down and then attack it. The altitude of the city is controllable and presumably it can come down to ground or at least very close since the city can get supplied without flying vehicles.

So how do you get it to come down?

The first method is to use the Greek approach. "Beware of Greeks bearing gifts" was not really about the Trojan horse. It was about the annoying habit the Greeks had of conquering fortifications by simply bribing a commander to open the gates or let them over the wall.

This could be easily adapted to your scenario. Get a messenger inside and promise to make a dozen men fed up with, let us be honest here, futile and suicidal resistance rich. You can even promise to spare the city atrocities and looting, the implication being that "If you refuse to become filthy rich there will be atrocities."

The second method is assassination. A resistance such as this has one or two moral leaders. Remove those and negotiations for surrender will be imminent. At the very least the morale will collapse to the point where offers of bribes will become very attractive. I mean, those people were killed inside the "unassailable" city, so nobody will be feeling safe. Leaving a letter on the bedside table with dagger on top of it has sometimes been enough. Sometimes showing you could assassinate is better than actually doing it.

So how do you get messengers or assassins in?

Of solutions proposed in other answers, hot air balloons would probably work. More epic and less tech level busting solution would be to wait for strong wind and use large kites. The prosaic, boring, and practical solution is to simply get your agents into the city before the siege. I mean seriously, you have known you will want to conquer this city for years is there any reason not to be prepared in advance? Especially when it is obvious that getting into the city after the siege starts will be a major pain.

You can either bribe necessary amount of soldiers in advance, there are no absolutely loyal troops with enough time to prepare, or infiltrate your assassins among the "invisible" servants or common soldiers. Probably both.

But seriously a floating city with only loyal soldiers inside is pointless. A city is its people. A fully evacuated city can be simply ignored until the doomsday comes and hell freezes over. Or the defenders get bored of pointless resistance ignored by even their enemy.

It would probably be possible(ish) but I'd pick something else than a ballista for stage one.

Use some siege weapon more adapted to hurling heavy payloads, like a trebuchet, for the first stage. Then make a projectile for the trebuchet of a fin-stabilized ballista with some sort of delay mechanism that triggers stage two...

Ok, it's not really practical, but not impossible and follows the rule of cool.

On second thought, you can use the trailed rope to stabilize the ballista and skip the fins.

• I haven't ruled a trebuchet out yet as the stage one, I'm just worried that the launched projectile would tumble once released. Maybe have the trebuchet mechanism pull the balista on a guide rail? – Lu22 Mar 10 '17 at 12:13

Hmm. I might have several possibilities for you. But bear in mind that I'm kind of an organic writer (I love coming up with these things then shooting them down one by one, or bettering one or two until I come to either a proper conclusion, or know what just isn't happening).

So let's agree on the limitations on which these possibilities will be based.

• No gunpowder
• No magic (or so little it might as well be none)
• No Deus Ex Machina
• No traitors or convenient serendipity
• Low tech

Right. So given you want a mechanical solution, let's see what this twisted mind of mine comes up with.

1. The glider option isn't impossible, but you're over complicating it. Your army is on open plains, but you are also looking at a literal army that needs to storm the gates. Getting everyone a glider is expensive. Getting all of them up in the air is nearly impossible. You need to take into account that some will have a fear of heights, or will be nervous and butterfinger-y, which could lead to them literally falling out of the sky. Given the era, safety regulations will be unheard of, so people will die on the climb regardless. So get into the meat of the theory behind gliders, set them upwind, and have them armed with crossbows with rope to make sure they can get into the city. This method guarantees heavy casualties, because it sounds like this city is heavily fortified. A castle-like structure. Heavy losses on the attackers' side is all but guaranteed, so it might be written off as acceptable losses. Still, the glider thing feels stilted, at best.

2. Why not go for the undercover route? I mean, this place is hovering, why not just use ballistas to attach ropes to the underbelly of the city and have your people dig up until they're in? You could have them attach the ropes in the dead of night, to ensure they aren't under constant arrow-fire. But this route assumes it's doable--very dependent on there being no handwavium or super dense adamantine flooring.

3. I've considered the balloon thing, but that's a slaughter waiting to happen. I mean, if the defenders have ballistas or bows and a plethora of arrows? Balloons aren't the most defensible places to stage an attack. What the attackers could do is have a portion of their army up in balloons with either ballistas or crossbow, or whatever, and rain down cover-fire on the defenders while their compatriots are scaling the walls. But this means knowing well in advance what the winds will be like, and so much hair-thin precision planning that it would take a genius (or plot armour) for it to work. It's not impossible, though. And if you have a character that's already shown themselves the tactician, this is the better route. Still, much room for error, and that isn't always something you want. And that's ignoring the ginormous material costs to this approach.

4. Sieges are about starving the enemy. You know they cannot leave without your notice, and they are essentially trapped. So make them miserable. It's boring as all hell, but why not smoke them out? I mean, you're in a plains. That means the wind is typically predictable, so you can plan for where the fire needs to be lit. There won't be any air purifiers on the defenders' side, and this would essentially kill them slowly (though less slowly than starvation). If you have a chemist on hand, or a woodsman, they might know of local flora that might add a poisonous substance to the smoke cloud. Then all you need to do is wait the little bastards out.

5. If taking this place is imperative, and you need to keep the place intact, then you need to get creative. Look at the people in the attacking army. Who has the skills needed to get this job done. Are there any that were once of the rogue-archtype? A thief, a pickpocket. Something, anything. These types tend to know how to get around in small groups. Get them what they need--a long rope to climb up, that glider to swoop in, parachutes made of silk to paratrooper their way in. Get them in and let them raise hell among the enemy troops. Start fires in the enemy food stores. Spread rumours about a black magician, or a necromancer that will raise their corpses to ravage their loved ones.

That's all I can come up with. I need coffee (preferably a non-poisoned variant).

## Use thermals to get your glider up to height.

The dual glider approach is impractical because the launch mechanisms for flinging the second glider will necessitate a massive first glider, and an even more massive launch apparatus to get it up to a decent height. Instead, your siege engineers should focus on figuring out how to get your first glider high enough to harpoon the city on its own.

A good way of doing this is to use thermals. Thermals are rising columns of air that gliders or birds can circle in to gain altitude. Modern glider pilots of all forms use these to prolong their flights. You can ground launch your glider, using some sort of ballista-like mechanical assist, and then use thermals to climb to a sufficient height before harpooning the city. If there aren't enough naturally occurring thermals in the area, your troops could build a large bonfire to create one artificially.

Dig a big hole under the Eastern half of the city. Pile the dirt up under the Western half. Pretty soon the height differential causes a list to starboard. Continue until the city capsizes or everyone falls off, whichever comes first.

Or for something a bit more evil and less daft, how about burning the town to ash with an Archimedes heat ray? Or if the city is near weapons range, firing dead animals, beehives, and the severed heads of captured enemies at them (to spread disease and generally prove your evilness).

How do people normally get into the fortress city? The Empire will do that.

A lot of castles protection was offered by inaccessibility, your city has this in spades, yet there are no historical examples that I know of of people firing ropes up the walls. Mostly they used siege ladders and siege towers.

Most Castles that were assaulted by an army and taken by force had a traitor do the dirty work and let them in.

Most of the time castles were besieged and lost that way, you have suggested that they send away all non combatants to preserve food stock, this is a good thing for them, and a valid tactic, but as this is not actually a castle but a nation-city is it reasonable? Could the empire not capture enough of the citizens and force a surrender that way?

If you are looking for a mechanical solution then I think you will struggle, mainly with the idea that even if they somehow get a rope up to top of the walls, the attackers can climb this rope without any of the defenders simply unhooking it.

• Unhooking it or throwing boiling oil or water down the line, then unhooking it. – Wtrmute Mar 10 '17 at 17:49

The other answers are excellent, so I just wanted to address the problems with your "firing a ballista from a glider" solution. You're neglecting Newton's Third Law:

For every action, there is an equal and opposite reaction.

There's a reason large siege engines were firmly secured to the ground and usually very heavy. If you want to toss a projectile one direction, you have to have something to push against.

A glider is attached to nothing (more or less -- it's complicated), so unless the glider is very heavy much of the kinetic energy that you expect to drive the bolt forward will actually push the glider backward. It might still work, but you'd have to get fairly close to the city and the bolt will have much less momentum than you might expect.

Ah, but you say, the ballista will fire upwards at an angle and not forward in the direction the glider is moving! Ah, well, you see in this case all of the opposite kinetic energy will be directed down into the (presumably) lightweight and fragile air frame of the glider. Not a pleasant ending for your hapless aeronauts.

But, ok. Let's say you use balloons instead. You still have to factor the weight of the rope itself. Rope that someone can actually climb (without special tools) would have to be fairly thick -- think the 2" manila rope you find in gym classes. This rope is about 1 lb per foot, so if the city floats 500 feet above the ground (not particularly high), that's 500 lbs of rope. Not impossible, but not insignificant, and certainly not something you would attach to the end of a bolt.

Instead you attach some kind of lightweight rope through the "eye" of the bolt (like a needle), doubling it so you can keep hold of the free end. Then you attach the lightweight rope to the heavier rope and pull it through until the heavy rope is near the edge of the city or wherever it's possible climb up to.

There's still the problem of how to secure the rope, and whether the weight of the climbing rope (plus the climbers) wouldn't detach the bolt, but you get the idea.

So what is the solution? Others have proposed various things, but I think infiltration is most obvious, especially if those in the floating city don't know about balloons and other ways to float non-magically. In the dead of night, send a few dozen guys up by balloon, with ropes, and have them attach the ropes at various points.

Heavy rope? Not a problem if you also give your climbers ascenders of some kind. Then you can use a much lighter weight rope. These can be fabricated out of anything, although of course if they're too flimsy they could be unreliable.

• Why not send in infiltrators with the food supplies that the city brings in, hidden amongst shipments who then come later to put down ropes. They could hide in barrels of salted meat, fish etc. – Sarriesfan Mar 10 '17 at 19:17
• @Sarriesfan sure. After all it's a story and anything the author wants to work can work. But there are two problems -- 1) presumably the leaders of a floating city know their food supply is their weak point, and guard it appropriately, and 2) if the city is already under seige then they wouldn't be taking on new supplies. So you have to bring a bunch of guys in early and have them hang out in the city for days or weeks, each carrying a big coil of rope. Suspicious? – Andrew Mar 10 '17 at 19:28
• supplies, what supplies?, resupplying would be nearly impossible on a open plain any siege worth its salt could see supplies coming from a long way off and just capture them. plus the city lowering ropes and descending would be advertising incoming supplies. – John Mar 10 '17 at 20:45
• @Take a play from the Greeks at Troy but slightly modify it, lay a siege for a time making the city short of supplies. Break that siege and the city may be desperate enough to resupply it's army and citizens that it is sloppy in its food security measures. Plant your infiltrators and then march back at night to attack the city. – Sarriesfan Mar 10 '17 at 21:27
• @Sarriesfan I'm not sure if OP's Evil Emperor has 10 years to wait, but again, it's not my story. – Andrew Mar 10 '17 at 22:15

Problems with the multi-stage ballista

1. Structural stresses on the glider.

The glider would be subject to a very hard acceleration during launch. In many ways, this situation is similar to catapult launching of aircraft which puts 3.6 Gs on the pilot and airframe.

https://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Acceleration.pdf

1. Energy spent lifting the second stage.

While momentum transfer in ballistics does degrade as you approach the extremes of mass. I.e. something that launches a 1kg object at 10m/s can't inherently launch a 1g object at 10km/s, Overall, any energy you spend lifting that second ballista is not going towards lifting the final bolt.

1. Newton's 3rd law.

Without the ability to meaningfully brace, the glider is going to absorb much more of the stored energy that what you would want.

1. It'll be an easy target

This ballista would have to be very large, most likely constructed in place, within easy view of the besieged city, which can let gravity work for it.

Possible Alternatives

Building a raised launch platform or a ramp to the city such as the Roman siege of Masada.

http://www.biblicalarchaeology.org/daily/biblical-sites-places/biblical-archaeology-sites/the-masada-siege/

While this would be time consuming, and workers would be exposed to fire, you could eventually build up to the city, or simply raise your catapults high enough to hit the city.

Your Empire could also target the displaced civilian population. Eventually the Empire will probably want to claim these neighboring states as well anyway. Attacking when they are forced to deal with large refugee populations, would be opportune. Once you have hundreds if not thousands of hostages you are easily in a position to demand the surrender of the city.

A risky approach would be to seemingly abandon the siege, but leave a contingent camped directly underneath. This contingent would be nearly invisible to the city above them. Eventually, when a city's scout descends to look around, it might be possible to send a strike team up the same way, allowing the breach necessary.

or, stay with me here, or the gliders could just have a fuse that burns all the way to the main charge once these things should be at their highest point.

Here are a couple of things that people literally always forget.

1. People have known how simple kinematics work for a very long time so any civilization with a level of technology around that of ancient Greece should be able to make easy calculations like this.

2. Gunpowder might not have been around for as long as kinematics has, but it's not new. There are a few scenarios that would make it possible for a civilization at the tech level of ancient Greece to get it from a more advanced civilization who simply over looked it.

These two points bring me to the second part of my answer. Why not just mount the ballistas on a siege tower? Alexander the Great had siege towers. If I'm not mistaken - I didn't look this up so I probably am at least somewhat wrong about the following statement - the Romans had and used siege towers. Can a ballista hit this floating city if it fired from 60ft in the air?

Double ballista sounds cool, but there are just better alternatives available using period correct technology.

tl;dr ancient people are smart likely could have laid siege to such a city with their technology without double balista.