The question is about stopping arrows fired as a barrage by an army far away. You control fire, and suddenly blast the entire barrage of arrows with fire hot enough to incinerate the wood in a matter of seconds. Would the not-completely-molten arrowheads still be able to roughly reach the target (a moving band of soldiers)? Or would the destruction of the shaft, partial melting of the arrowheads, and force from the hot air (fire is basically hot gas particles that glow) be enough to change their course completely?
Arrow heads are undamaged, shafts are slightly blackened - but they're going to miss for 3 reasons:
In extreme cases the winds from fires can reach 150 mile per hour winds. That's 241kmph. Recurve bows fire at about this speed. You're not going to stop it mid air in time, but you can throw it off course by adding a 240kmph cross wind.
Even if the archers can predict the updraft and adjust their trajectory, the wind from the fire isn't uniform, and the arrows moving from one region to another with vastly different wind speeds will ruin their trajectory. A wind shear of about 30mph (ie 30mph difference in wind speed between two adjacent regions) has downed a commercial airliner. Wind shears from fire can get strong enough to make fire tornados, which make arrow trajectories tricky to predict.
Fin / feather melting
These delicate fins are very susceptible to fire:
The wood will partially protect one fin more than the other(s), meaning they'll melt unevenly. When these melt unevenly, it will throw off the aerodynamics. The arrows will turn and start tumbling. When they tumble they'll slow down and fall short. If they do reach you - you will probably be hit by the arrows side-on, it'll be like a soft caning rather than an arrow impact.
Yes, but you don't need anywhere near that much heat
Enough heat to incinerate the shafts might melt the heads, but it's irrelevant. Without the shafts the heads will begin to tumble and they'll fall far short of their targets.
But you don't need anywhere near that much heat. Those arrows were aimed given the archer's best estimate of the meteorological conditions at the moment of release — and you just changed them. Heat rises. All you need is enough heat to materially effect the arrow's trajectories and they'll still fall short. Such a blast of heat might be more believable than a blast so devastating that it would be more useful to use it against the archers themselves rather than their arrows.
Such a blast would be the fantasy-medieval equivalent of a nuclear bomb.
Unless you manage to completely destroy the arrow shafts, you have a problem.
The feathers help with accuracy, but the aeroynamics of a heavy head and lighter shaft will still keep them pointed in the same basic direction.
It takes little fire to set a wooden shaft on fire. It takes a HUGE amount of fire to completely destroy a wooden shaft in 2-3 seconds! (try cutting through a wooden broomstick with a oxy-acetelene torch, to test. It takes about 30-40 seconds! You can cut through a mild steel bar of same diameter faster than wood)
So you have a barrage of incoming arrows. You have heated the arrowheads to scorching heat, you have blackened and set on fire the wooden shafts. You have destroyed the feathers. You have nudged the arrows a bit off path due to winds from your fire.
Net result: You have made the barrage MORE dangerous. It is a barrage. Not accurate pinpoint fire. If the arrows arrive anywhere in the direction of the target, they have done their job. Barrages of arrows is the medieval equivalent of saturation bombing.
BUT! The incoming arrows are now not just pointy objects that might penetrate your armor and defenders. It is now an equal number of ON FIRE, with superheated metal tip, pointy objects that are landing on your armor and defenders.
What hurts more, being hit by an arrow, or being hit equally hard by a hot arrow that is on fire?
If your fire IS hot enough to burn away the wooden shafts, then the arrowheads are now red-to-white hot blobs of molten metal. I would rather be pierced by an arrow, than have a redhot blob of molten steel land down my pants.
Hint: set the archers on fire, not their arrows.
My father was a fireman. Wood catches fire fairly easily, but it takes time to burn. I don't think that any temperature that fire can reach will burn the arrow shafts in time.
One bit of wisdom he imparted was that in a building fire, steel stairways soften and fail before old-fashioned wooden stairways could burn through.
The destruction of the shaft, and the feathers on it, will destabilise the arrowheads, and remove much of the mass of each arrow. They'll go off course and start to tumble.
They'll end up scattered and few, if any, of them will hit their targets. They'll also be going slower because of the tumbling. While being hit by them won't be pleasant, they've lost much of their momentum and will do far less harm.
The accepted answer is along the right line of thinking, but way understated. When you heat air up very quickly as you do in this scenario, the air will expand with explosive force. Heating air up enough to burn an arrow mid flight means you are creating a LOT of explosive force. Your fire wall would would closely resemble the mechanics of the Iron Curtain countermeasure that the US military uses for stopping missiles, but arrows have much less kinetic energy for their cross-section than missiles do, and your firewall will have much more explosive force than an Iron Curtain to be able to burn away the wood and melt arrow heads.
Instead of just blowing the arrows off course, you will blast the red-hot arrowheads backwards in a spray of shrapnel... and possibly burn both armies to death in the process...
I am no metal engineer, but I think temperature that is not high enough wouldn't do anything to metal arrowheads.
After quick search, wood start burning at 300 degrees (here ) and forging temperature for metals (alloys, but I don't think iron would be much further from them) is way higher then 300 degrees (here). Therefore it wouldn't make any difference.
That's a very general question, because of course it depends on the speed and design of the arrow, the mothod of firing the arrow, etc. Already we have incendiary rounds and tracer bullets that have no problem reaching targets while on fire; a red hot piece of metal is not melted (and isn't close to melting range yet for steel) so conceivably there wouldn't be a problem with sending hot arrows.