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What if these weapons existed before the start of the war?
http://en.wikipedia.org/wiki/Bazooka
Apparently Robert Goddard had successfully completed a similar weapon before the end of WW1, but the armistice was signed before it could enter manufacture.

So what if the US or Germany had worked on such a weapon in the 20's/30's and had a functional unit at the start of the war that was ready for widespread use?
Let's make a distinction though: the Bazooka and Panzerschreck were RPG's, while the Panzerfaust was a recoilless AT weapon. Both technologies existed before WW2 and were very simple to implement, so that obstacle shouldn't derail this WI.
Still, how could either the Germans or US justify the production of such a weapon before the war?
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