kopfüber
headfirst
With the head first or pointing downward.
Er sprang kopfüber ins kalte Wasser.
He jumped headfirst into the cold water.

headfirst / upside down
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headfirst
With the head first or pointing downward.
Er sprang kopfüber ins kalte Wasser.
He jumped headfirst into the cold water.

The word is a compound of 'Kopf' (head) and 'über' (over). So it literally describes a movement in which the head is over the rest of the body or goes first.
Das Kind hing kopfüber an der Kletterstange.
The child was hanging upside down from the climbing bar.
As an adverb, 'kopfüber' (headfirst/upside down) describes the manner of an action. It therefore usually stands directly after the conjugated verb or near the verb it modifies.
Er stürzte sich kopfüber in die Arbeit.
He threw himself headlong into the work.
In its literal meaning, 'kopfüber' (headfirst/upside down) is often combined with verbs of movement. Typical examples are 'springen' (to jump), 'fallen' (to fall), 'stürzen' (to plunge/fall), or 'hängen' (to hang).
Der Akrobat ließ sich kopfüber vom Trapez fallen.
The acrobat let himself fall headfirst from the trapeze.
You can recognize the figurative meaning from the verb. When 'kopfüber' (headlong) is used with 'sich in etwas stürzen' (to throw oneself into something) or 'sich in etwas begeben' (to enter into something), it usually refers to a rushed, unconsidered action.
Er hat sich kopfüber in das neue Projekt gestürzt, ohne die Risiken zu bedenken.
He threw himself headlong into the new project without considering the risks.
hasty
'hastig' and 'kopfüber' can both describe someone doing something very quickly and without enough calm. 'hastig' fits especially when an action is hurried and therefore often seems imprecise or restless. In this relation, 'kopfüber' means more that someone throws themselves into something without thinking about it for long. It sounds more figurative and stronger. For a quick movement or task, 'hastig' is often more natural; for a rushed decision or a step taken without much thought, 'kopfüber' fits better.

Nach dem ersten Gespräch stürzte sie sich kopfüber in das neue Projekt.
After the first conversation, she threw herself headlong into the new project.
Vor dem Termin schrieb ich hastig noch eine kurze Nachricht.
Before the appointment, I hastily wrote a quick message.