1. **State the problem:** An airplane is flying due south at 500 km/h and encounters a headwind from the south at 40 km/h. We need to find the resultant ground velocity of the airplane. 2. **Understand the vectors:** The airplane's velocity vector is $\vec{v}_a = 500$ km/h south. The wind's velocity vector is $\vec{v}_w = 40$ km/h from the south, which means it blows north (opposite direction to airplane's motion). 3. **Calculate the resultant velocity:** Since the wind opposes the airplane's motion, subtract the wind speed from the airplane's airspeed: $$\vec{v}_r = 500 - 40 = 460 \text{ km/h south}$$ 4. **Interpretation:** The airplane's ground speed is reduced by the wind speed because the wind blows against it. **Final answer:** The resultant ground velocity of the airplane is $460$ km/h due south.