pel

The root pel comes from Latin pellere, "to push, drive, or strike" — the action of shoving something so it moves whether it wants to or not. Its past participle was pulsus, which is why half the family is spelled pel- (the live verb) and the other half puls- (the result). Picture a single shove. Then watch a prefix point that shove in a direction, and you get almost the whole family:

- ex- (out) + pellere → expel: drive someone out (out of school, out of the country, out of your lungs).
- re- (back) + pellere → repel: push back — an army repels attackers, and figuratively a smell or a person pushes you back, i.e. disgusts you.
- pro- (forward) + pellere → propel: push forward — exactly what a propeller and rocket propulsion do.
- in-/im- (in, on) + pellere → impel: push someone into action, an inner shove. The noun impulse is that single push felt from inside — a sudden urge to act before thinking.
- com- (together, intensifier) + pellere → compel: push with full force until there's no choice — to force. Compulsory schooling is schooling you are pushed into; a compulsion is an inner push you can't resist.
- dis- (apart) + pellere → dispel: drive apart and away — you dispel doubts and fears, scattering them.

The puls- branch keeps the literal beat of pushing. A pulse is the heart pushing blood, felt as a throb; to pulsate is to keep beating. Every -sion noun (expulsion, propulsion, repulsion, compulsion) is just the puls- form of its verb.

One corner of the family takes a detour through French. Latin ad- (toward) + pellere gave appellare, "to address or call upon someone" — to push your words toward a person. From there English got appeal (to call earnestly upon someone, then to call upon a higher court) and the legal terms appellate, appellant, plus appellation (the name you call something). Repeal is re- + appeal — to call something back, i.e. to cancel a law. The connection to "push" is faint here, but the thread is the same: directing force or a call at someone.

The rule of the family: pel/puls is always a shove. The prefix tells you which way it goes — out, back, forward, in, or together — and puls- is just the spelling that shows up in the noun.