A TRACON Departure Controller reported a departing aircraft flew below the Minimum Vectoring Altitude after they became confused by ATC instructions and deviated from their assigned heading.

Narrative

Aircraft X departed Runway XX with a destination to ZZZZ which is northeast of our airport. To comply with Opposite Direction Procedures; I turned the aircraft left heading 080 and asked for their on course heading to ZZZZZ; the first fix. I had every reason to believe that aircraft performance characteristics would allow the pilot to make this turn within the roughly 2.5 miles between the departure end of the runway and the edge of the 6;000 foot MVA. When I asked what the on course heading was; the pilot responded that they were on course. I told the pilot negative; they were not cleared on course and to fly heading 080. I asked again what the on course heading was. At this time the pilot became confused and appeared to have stopped their turn; leveled at 4;500 and was now heading south towards the MVA.Moments later; Local Control called to inform me to watch for the MVA and I responded that I was. I told the pilot multiple times to expedite their turn and climb. The plane then entered the edge of the MVA at 5;300 for a moment before turning back north. On the part of the Controller; I feel that I could have told the pilot to turn left heading 080 upon reaching 6;000 but this is assuming that I felt that the plane couldn't make the turn. I also could/should have issued a safety alert even though the advisory never appeared. As far as procedures; I believe that there is already work being done to update the airspace maps with a possible shelf being added in this area.