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Imagine a medieval-style city, besieged by enemy army. It might be big and have wide river preventing besiegers from direct attack on the walls and launching projectiles significantly far into the city. Besiegers want to use hot air baloons to drop burning projectiles and biological weapons in form of infected meat etc. as well as drop some attackers during nighttime to open the gates of the city. Is it possible for hot air baloons to perform such missions?

Better for defense.

I pondered this scenario in 2009 and posted the same idea.

https://www.halfbakery.com/idea/Siegebreaker_20balloon#1249939283

The excellent comment by Bunsen Honeydew points out that the tethered balloon scheme has more advantages when carried out by the defenders within the city as opposed to those outside laying siege.

Consider: the besieged are at the center, and the besieging are at
the circumference of a(n idealised) circle. The balloon is anchored in
the city, and the anchor point is surrounded by all the infrastructure
to support it - furnaces with hot air/gas hoses, ammunition supplies,
bucket brigades, pipework, extra fuel, spare parts, winches,
windlasses, mules, encamped technicians, support staff for the support
staff, ancilliary cooks, batmen and wenches, and so forth. Should the
balloon need to replenish itself of any vital supplies, it can be
winched back in to the center.

Now consider a change in the winds: the balloon changes tack, but is still (given a few moments of belaying or withdrawing the
tether) hovered above the enemy.

In the inverse case, where the circumferential siege-layers are attacking the central hold-fasts, a change of wind immediately places
the weapons platform off-target. In order to reaquire the target, the
anchor point - and all the associated infrastructure - must be moved a
great distance across the landscape around the circumference, or one
must wait until the wind is once more in one's favour.

Insert a river, a forest or other obstacle anywhere about the city and the case for the attacking forces becomes particularly dire.

I hold, then, sir/madam or other, that this system is inherently defensive, rather than offensive, in the siege situation.

— BunsenHoneydew, Aug 04 2009

My initial thoughts are that hot air balloons would be very vulnerable to projectile weapons like cannons and arrows. As these balloons would be primitive, I am not sure how high they could fly or how well they could be maneuver over the target. Another issue is payload capacity. Most of the balloon's floor would be used to burn something in order to create hot gas, that will lift the balloon into the air. Some more space will be occupied by the pilot and perhaps the bombardier. There is a good chance that anything that was on fire and dropped from a great height, would extinguish itself before it stuck the target on the ground. That is assuming that objects could be reliably targeted to begin with. These biological agents would probably have just as much chance to infect the pilots as the targets on the ground. In order to drop of troops, said balloon would have to either land or fly super close to the ground. The light generated by the fire would give away the balloons presence. I think a trebuchet could accomplish most of want you want to do, with out having to actually leave the ground.