the first big thing is that the wire is stored in the missile, not on the launcher, meaning it is just leaving wire behind, not pulling it along, which is much less stressful on the wire.
Second, the wire is reeled out around the exhaust and not directly into the engine of the missile, to minimize the temperature exposure, and then because the missile is traveling so fast any wire that is exposed to the heat is only exposed for a split second before it is reeled out and left behind by the missile.
as for maneuvering, this is an actual concern and different missiles only have so much ability to maneuver. They can only make turns at a certain angle maximum. most of this issue is avoided by, well, not launching the missile till you are pointed mostly at the target and then only having to make small adjustments as it flies.
Because they don’t put any pressure on them.
Take the TOW missile. It has two wire spools on the missile itself (sufficiently separated to the side from the rocket motor that the wire doesn’t end up inside the thrust jet) and as the missile flies the spools distribute more and more super-thin wire. The wire only has to be strong enough to drag out more wire from the spool (which is of course designed so that there is minimal resistance).
The reason why there are two spools is to make the drag/weight on the missile symmetrical (the wire does pull slightly and weighs a bit) and two wires mean that if one wire breaks the other can still send the control signal.
The TOW as an example separates the rocket exhaust from the wires. The wires leave from the rear of the missile body and the exhaust exits from the middle (on the sides). The wires on the other hand leave from the missile so the only force on them is what’s required to pull them off the spool.
From a friend that found a reel, the wires about just pop off it with a tiny tug.
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