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Bomb load was at about 7 tons for the A-5. the A-5 was capable of reaching 350 mph and even more if using a shallow dive (as they used to do on bombing missions agains the UK)
If You trace that 350 mp/h figure down it comes from a He-177 Schiffer book. Those Schiffer books have never been reputated for reliability altough in this very topic they were correct. However, there is something different:
The 350 mp/h were flown at "Erhöhte Notleistung"-powersetting (something like WEP + injection. Only suitable for less than a minute in DB 610´s without risking an engine failure or worser: engine fire) in level flight by an He-177AV prototype stripped down from defensive armement and without payload except for minimum fuel load. So this figure isn´t represantative for the He-177 in usual bomber configuartion.
The He-177A5, the mostly used bomber variant had a top speed of 440 Km/h (273 mp/h) sustainable for 30 min.+ at "Steig- und Kampfleistung"-powersetting (100%). Using the "Start- und Notleistungs"- powersetting (WEP or 110%) the He-177A5 could be accelerated to 488 Km/h (303 mp/h) max in level flight. This speed was flown at Rechlin during several testflights with defensive armement but without bombs at 3/4 fuel load (for normal range).
Hope, this helps.