*(s)p(h)er-

The core idea of this root was:

strewing, spraying, scattering in small pieces, sowing

Think of blowing on a dandelion in its white stage, or slapping the hand onto a REALLY dusty old sofa.
The original sense is pretty visible in words like to spray, sprout, spurt, sprawl and spritz. 

Here’s an incomplete list of the English family members:

• to spray 
• spritz (import from German “spritzen”)
• to spread (focused on “distribution”, after a burst, cousin of German “spreizen” which is to straddle)
• to sprout (variation of “spread”, in the sense of growing)
•  spore (“a small seed spreading”)
• sperm (“what is sown”, used to refer also to seeds in ancient Greek)
• sparse (“spread out” with a focus on lack)
• disperse (“distribute, spread out” with a focus on low density)
• intersperse (“sprinkle/strew in between”)