In your AnswerCorrectWorld world class code, the 'if' statement (lines 23 through 26) should not be in the constructor of the world, but in an 'act' method for the world.
In your BattleArrow actor class, you need to get the value of 'gameWorld' from the currently running BattleWorld world. To get that value, use;
This is the world that you need to pass along. BTW, I am assuming that BattleArrow objects are only instantiated from within the BattleWorld world and never used in any other.
BattleWorld bw=(BattleWorld)getWorld(); World gw=bw.gameWorld;