The Spanish empujar (“to push”) has the same common ancestor as the English for the same, push: the Latin pulsare.
Pulsare meant, in Latin, “to beat”. A push is a sort of beat, in both senses: a punch and, a punch happening over and over again!
From the same root we also get the English, to pulse, of course. As does… impulse. Yes, an impulse is indeed a strong punch!
The sound here is a variation of the sh-to‑j pattern, where variations of the s/x/sh/soft‑g sound in Latin turned into the “j” in Spanish (via the Arabic influence) but remained the same as it transformed from Latin into educated English. Hence the “sh” sound in “push”!
Quejar, Spanish for “to complain” doesn’t seem related to any English equivalent.
But upon closer look, it is a first cousin of both quash and squash.
All come from the Latin quassare, meaning, “to shatter.”
The relationship is easy to see if we remember that the Spanish ‑j- sound used to be the Latin ‑s- sound (and many variants, like ‑ss‑, ‑si‑, ‑sy‑, ‑sh‑, ‑ch‑, etc).
Thus, the qu‑j for quejar maps to the qu-sh of quash and the sq-sh of squash.
Complaining, it seems, is a form of quashing (squashing?) your opponent!
Embassy (and Ambassador) and its Spanish equivalent, Embajada (and Embajador), both come from the same ancestor, the Old French Ambactos.
What is most interesting about these two is that it is an example of the pattern where the ‑j- sound in Spanish maps to the ‑sh- sound (and its cousins, like ‑ss- and ‑ch-) in English. Remember syrup and jarabe, chess and ajedrez, sherry and jerez, and push and empujar for a few examples.
Thus, the m‑b-j of emabajada maps to the m‑b-ss of embassy.
Jeringa, Spanish for Syringe, sounds like it has nothing in common with its English counter-part. But they are literally the same word.
The Latin sh- sound often evolved into the j- sound in Spanish — originally retaining the sh- sound but eventually, under Arabic’s influence, transforming to the throat-clearing sound we know and love.
This explains how both jeringa and syringe derive from the same root: the Latin siringa, itself from the Greek syringa. The sy- sound is a variation of the sh- sound and therefore the sy-r-n‑g of syringe maps to the j‑r-n‑g of jeringa.
The Spanish for syrup, jarabe, comes from the same root as the English: the Persian/Arabic sharab, which means “a drink, or wine”.
The drastically different (at least superficially) words are explained by the sh- and related (such as, sy- ) sounds changing to the Arabic-sounding j- sound in Spanish — but not English.
Thus, the j‑r-b of jarabe maps to the sy-r‑p of syrup.