Re: "120 yards", the rest of the post uses feet as the unit so this value should be in feet as well.

Since ballistae or contraptions like that are just bigger (cross)bows: do they also have the same speed (= upward range), just with heavier projectiles as well?

@GrumpyYoungMan "so this value should be in feet as well" - metres would be even better IMO - that way the majority of the planet could understand it :)

@GrumpyYoungMan the poster is just trying to be consistent, within the system they are using. They should have been talking about feet, for altitude the balloon is at. Yards, for how high the arrow can shoot. Fractions of a mile for allowable historical balloon heights, and fathoms for anything being dropped from the balloon. Worst of all, I'm being serious here! American standard dictates that those measurement units be used for those instances of measurement. The rest of the world is lazy, and uses meters to measure everything.

No really seriously. Americans standard specifies what units one is allowed to measure things in. A drag race is 1/4 mile. never 440 yards. never 1320 feet. never 2 furlongs. Even though "2 furlongs" is much easier in the tongue than "quarter mile".

@Hobbamok many ballista use rope torsion which works very differently than simple bending. they should have very different constraints.

Just as a note, it becomes POSSIBLE if you have a sufficiently built recurve bow or modern bow. A good compound bow today can hit 340 FPS (almost double your calculations), and some longbows can even push 210 FPS (which could pretty readily hit the 600 mark). This doesn't change the answer for bows and crossbows, but if you really needed to hit that balloon, there are a few variables affecting arrow speed and ways to do so.

I would also add that the current world record for Flight Archery is almost 500m. I'm not sure about the math to calculate the height needed, but with a recurve bow, the requisite height MAY be reached, wind speed's affect on the carry distance nonwithstanding.

@Anoplexian advertised fps by the manufacturer and actual fps are rarely the same., but yes modern cam bows can achieve higher velocities, of course if you are in the modern era just use a gun. Also trying to calculate velocity with distance gets very tricky unless you have the exact angle it was fired at and measurements for windage at multiple elevations wind, which is why I went for measured velocities, its just more accurate and accessible.

@AaronF but what medieval person is going to know what a meter is. Yards and feet however were the measurements used at the time. but I added all units to make everyone happy.

@John lol, good one :-D although it depends on the country in question (actually, after looking at this, I see that it depended on the region or town in many cases!) Thanks for the metric measures, though :) (what is 1000lb, mentioned as the weight of the bolt, btw? It sounds quite heavy!)

@PcMan - that is gross exaggeration. Yes, America requires pilots & air traffic control to specify altitude in feet. The rest of the world does too, because America got there first and it was too important for international flight safety to play political games about. There are also certain rules about what units government data should be published in, and in which data should be supplied to the government, But this done for consistency, and every government has similar rules. As long as you are not dealing with the government, you can measure in any units you like.

I know the OQ didn't ask about balistas, but they get about double that velocity, which means they'd theoretically be a bit more dangerous, but not much.

@AaronF bows and crossbows have a draw weight measurement, it refers to how much weight it takes to fully load (draw) the bow. using only weight.

@PcMan Just wanted to point out that as a native speaker in the United States (unfortunately), I've never heard someone actually use a furlong as measurement.

@ReinstateMonica3167040 yep. Americans measure land distance in miles, if its far, or in yards if it is close. People are measured in feet and inches, not just in inches(except babies, or genitals). Speed is miles per hour, if the speed is done on the ground or by a living being. On water or flying, it is knots. NB unless it is alive. A falcon flies at 60 mph, the microlight airplane that is pacing it flies at 52 knots. BUT! the airplane has a ground speed of 60 miles per hour. Each american measurement has a unit of measure that is reserved for that use, even when others are valid.

@ReinstateMonica3167040 I stuck furlongs in there to poke fun at everyone complaining about units.

"no air resistance" is a pretty big deal here. An arrow would realistically not exceed 80 vertical meters due to drag. Furthermore, it will lose a lot of its energy early in the arch because air resistance is stronger the faster you go. So, the odds of an arrow even being dangerous to targets more than about 40-50m up is pretty unlikely. desmos.com/calculator/on4xzwtdwz

@Nosajimiki is there a reason you halved the velocity and gave the projectile such a ridiculously high drag coefficient? you used the drag coefficient of a sphere, an arrow should be closer to 0.05-0.08. you also have diameter of the arrow set to almost half a meter. Fix your variables and you will find there is actually very little effect to air resistance. sites.google.com/site/technicalarchery/technical-discussions‌​-1/…