A combination of radio beams and that basic IR sensor. The radio beams for rough guidance and the IR sensor to ensure that it goes to the center of the city or wherever the most heat is.
Another issue that I can think of the IR sensor alone is its sensitivity at range, you’d need a lens assembly. Normal glass transmits IR fine especially the glass of that day as it was so primitive --almost no special formulations or coatings. So it is totally possible that a device that resembles a fly eye could be made. It would consist of a group of long focal length lenses that project onto the lead-sulfide sensors which would then guide the missile in.
You could only do this at night though as the ocean and ground will heat up during the day decreasing the temperature differential plus you can’t forget the Sun itself! Probably, the best time would be calculated by determining the max altitude of the missile, it’s max line of sight at that altitude so it won’t see the sun accidentally, and, the longest time after sunset so the ocean and ground around the city has had the most time to cool-off making the city the hot spot.
But, also, you could have it launched with the calculations regarding the rotation of the earth and have it get really close to the target. Once, it’s on a downward trajectory the sensors will turn on as they will be facing downward, which will alleviate the issue with the Sun a bit. Then, it can fine tune it’s trajectory and go for the center of the city.
Some technical issues with that technology of that time:
IR sensors, probably really slow to react which may be a good thing as it will smooth out any erroneous hot spots or flares.
Lenses: unless you have a lens hood on them you can get stray sunlight into them. In reality, you could design the missile to have a lens hood type assembly up front but you would have to compensate for how that hood would affect its aerodynamics. But, the missile does not need to be going Mach 10. Probably be sub-sonic any ways considering the technology of the time. Stabilizers up front and in the rear could keep it on track. Be odd looking though.
Electronics: could be primitive transistor based! According to wikipedia:
So, maybe some dark ops going on there made a few batches of transistorized missiles? But, valves/tubes are also practical and were far more developed in general. The tubes would be the heavier option due to the typical power requirements of heating them up and their grid voltages. But, they did have miniaturized versions and it only needs to last so long so longevity is not an issue.
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