As an English-speaker I would be tempted to translate "Jack jumps up and down" into Jack springt auf und ab, but the English use of "up and down" seems idiomatic and my literal translation might be totally wrong. How does "Jack jumps up and down" get translated? Are there different ways of saying this in German? I've also seen in die Luft (zu) springen, or ständig aufspringen which to my American ears seems the more "German" way of saying it.
The key to translating this idiom is using the verb hüpfen (to hop, to skip, to frisk) instead of the verb springen.
Jack jumps up and down.
Jack hüpft auf und ab.
This is exactly the same picture as the English expression. The alternative
Jack springt vor Freude. (or Jack hüpft vor Freude.)
is very formal und sounds right only in narration. Then usually in Präteritum tense of course.
(Note there's another verb hoppeln which also means to hop. That's what bunnies do, people with a leg injury, and cars with broken suspension. A third verb hopsen means to skip in its narrowest sense. Hüpfen is a catch-all.)