On purpose - purposely - purposeful

Someone's purpose is, according to OED (at meaning 1. a.): "That which a person sets out to do or attain; an object in view; a determined intention or aim." In other words, one's purpose is one's motive, or intention. When one does something with the intention of achieving a particular effect, one has acted on purpose. A child will be more severely admonished if it has hurt a sibling on purpose, rather than accidentally.

• The single adverb that carries the meaning of this adverbial phrase is purposely. OED defines this as "On purpose, by design; intentionally, deliberately" (AWE's emboldening).

This should not be confused with the adverb purposefully, nor the adjective purposeful.

• Purposefully is an adverb of manner - that is, it describes how an actor acts, "In a purposeful manner'.
• Purposeful is defined in two ways by OED:
  • • 1. Having a purpose or meaning; indicating or evincing purpose or plan; designed, intentional.

• • • 2. Having a definite purpose in mind; resolute.

So purposeful shows what someone looks like, the impression s/he gives of being determined, and set upon a particular goal. Purposely has less to do with impressions and appearances, but about doing something deliberately.