strike, hit, knock, pound, dozen
打 centers on forceful contact: striking, hitting, knocking, or pounding. The 'dozen' sense is a separate loanword application that does not share the same conceptual center.
to hit (someone); to strike; to beat
to hit; to strike; to knock; to beat; to punch; to slap; to tap; to bang; to clap; to pound
launch (of a rocket, satellite, etc.); lift-off; shooting off (fireworks); (fireworks) display
blow; shock; strike; damage
safe hit
(preparatory) meeting; briefing session; (making) advance arrangements
repeated blows; striking repeatedly; barrage
end; close; finish; truncation; discontinuance; closure; cancellation; breaking off; aborting
batting line-up; batting order
batter's box; one's turn at bat
batter
pinch-hitting; pinch hitter
runs batted in; RBI
home run
batting average
hitting (a ball with a bat, racket, golf club, etc.); batting; batted ball; struck ball
break in the deadlock
overthrow; defeat; bringing down; knockdown
breaking down; defeating; abolishing
driving; pounding in; shooting into; implantation; invasion
to drive in (a nail, stake, etc.); to hammer in
value; worth; merit
percussion; tapping; examining by percussion
heavy blow; thump; wallop; belt; whack
打 combines 扌 (hand) with 丁, which likely contributes the sound and may suggest a nail or striking action. The exact historical development of 丁's role is uncertain.
A hand 扌 gripping a nail-like 丁 shape: picture hammering a nail with your hand to strike, hit, or pound.
For ダ, imagine striking a drum with a loud 'da!' sound: da -> ダ, and your hand hits the drum.