bullet, twang, flip, snap
彈 centers on a sudden, forceful release of tension: a projectile like a bullet, or a sharp motion like a twang, flip, or snap. The meanings all share the idea of something being propelled or flicked out from a taut state.
彈 combines 弓 (bow) with 單, which likely contributes the sound and the idea of a single projectile. The character originally referred to shooting a pellet from a bow, hence the bullet and snapping senses.
A bow 弓 on the left and a complex shape 單 on the right: imagine pulling back a bowstring and releasing a single pellet with a sharp twang.
For ダン, picture a bullet fired from a bow with a loud 'dan!' sound: dan -> ダン, like the crack of a snap.