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The mesosphere (the part of the atmosphere above the stratosphere where temperatures increase with altitude) is generally above the altitude reachable by balloon. It begins somewhere around 50 km.

When did a rocket first reach the mesosphere?

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This was almost certainly the German V-2 rocket. According to Wikipedia, on the 3rd of October 1942, a V-2 reached an altitude of between 85 to 90 kilometers. The second, and the first true "sounding rocket," was James Van Allen's Aerobee:

The first instrument-carrying Aerobee was the A-5, launched on March 5, 1948 from White Sands, carrying instruments for cosmic radiation research, reaching an altitude of 117.5 km.

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