1680s, type of firework, from sky (n.) + rocket (n.2). The verb, in the figurative sense of "to rise abruptly and rapidly" (often with suggestion of 'and then explode and vanish'") is attested from 1895.
c. 1200, "a cloud," from Old Norse sky "cloud," from Proto-Germanic *skeujam "cloud, cloud cover" (source also of Old English sceo, Old Saxon scio "cloud, region of the clouds, sky;" Old High German scuwo, Old English scua, Old Norse skuggi "shadow;" Gothic skuggwa "mirror"), from PIE root *(s)keu- "to cover, conceal."
Meaning "upper regions of the air" is attested from c. 1300; replaced native heofon in this sense (see heaven). In Middle English, the word can still mean both "cloud" and "heaven," as still in the skies, originally "the clouds." Sky-high is from 1812; phrase the sky's the limit is attested from 1908. Sky-dive first recorded 1965; sky-writing is from 1922.
[self-propelling projectile] 1610s, "projectile consisting of a cylindrical tube of pasteboard filled with flammable or explosive matter," from Italian rocchetto "a rocket," literally "a bobbin," diminutive of rocca "a distaff," so called because of cylindrical shape. The Italian word probably is from a Germanic source (compare Old High German rocko "distaff," Middle Dutch rokke, Old Norse rokkr), from Proto-Germanic *rukkon- (from PIE root *rug- "fabric, spun yarn").
Originally of fireworks rockets, the meaning "device propelled by a rocket engine" is recorded by 1919 (Goddard); rocket-ship in the space-travel sense is attested from February 1927 ("Popular Science"); earlier as a type of naval warship firing projectiles. Rocket science in the figurative sense of "difficult, complex process or topic" is attested by 1985; rocket scientist is from 1952.
That such a feat is considered within the range of possibility is evidenced by the activities of scientists in Europe as well as in America. Two of them, Prof. Herman Oberth and Dr. Franz Hoeff, of Vienna, are constructing a five-ton rocket ship in which they hope to reach the moon in two days. [Popular Science, February 1927]
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Definitions of skyrocket from WordNet
propels bright light high in the sky, or used to propel a lifesaving line or harpoon;