The core themes of this root were:
prospering, thriving, going forward
The notion of prospering is still alive in the words around the Latin branch of prosper (to prosper, prosperous, prosperity) and also in the English to spare.
The sense of going forward is visible in speed.
Last but not least, the root is most likely also the origin of despair and desparate which are based on the Latin sperare (to hope). So despair is essentially about losing hope (for prosperity).