Nice one Bultaco!
Clonemole
It is to do with manoeuvre margin. If you are slower than V2 + 15kts you do not have the required overbank protection, hence if you have an engine failure on takeoff your initial speed target is V2 to V2 + 20kts. If you are at anything below V2 + 15kts then you do not have the required overbank protection and hence the 'blanket' rule that any initial turn is limited to 15 degrees AOB. Here is what the FCTM says;
Immediate Turn after Takeoff - All Engines
Obstacle clearance, noise abatement, or departure procedures may require an
immediate turn after takeoff. Initiate the turn at the appropriate altitude (normally at least 400 feet AGL) and maintain V2 + 15 to V2 + 25 with takeoff flaps.
Note: A maximum bank angle of 30° is permitted at V2 + 15 knots with takeoff flaps. After completing the turn, and at or above flap retraction altitude, accelerate and retract flaps while climbing.
Immediate Turn after Takeoff - One Engine Inoperative
Obstacle clearance or departure procedures may require a special engine out
departure procedure. If an immediate turn is required, initiate the turn at the
appropriate altitude (normally at least 400 feet AGL). Maintain V2 to V2 + 15
with takeoff flaps while manoeuvring.
Note: Limit bank angle to 15° until V2 + 15 knots. Bank angles up to 30° are permitted at V2 + 15 knots with takeoff flaps.
If you look in the FCTM there are graphs which show the Manoeuvre Margin to Stick Shaker and you can see how the various flap settings and speed keep you on the safe side! The following is from the narrative related to these graphs;
When reviewing the manoeuvre margin illustrations, note that:
• there is a direct correlation between bank angle and load factor (G's) in level, constant speed flight. For example, 1.1G corresponds to 25° of bank, 1.3G ~ 40°, 2.0G's ~ 60°
I believe it is a certification requirement for a 15 degree overbank margin when in a 30 degree AOB turn.
Hope this helps and good luck with the course!
PP