It would not be possible for the timpanist to play a sustained note and simultaneously change the pitch of another drum without using a set of pedal timpani.
However, early drums did not resonate nearly as long as modern timpani, so composers often wrote a note when the timpanist was to hit the drum without concern of sustain.
However, the choice of stick during a performance is entirely subjective and depends on the timpanist's own preference and occasionally the wishes of the conductor.
