2D Relative Motion: The Resultant Vector

When an airplane flies, it experiences motion similar to a boat crossing a river. The wind acts like a river current, affecting the airplane's path. To understand where the airplane actually goes, we think about two motions happening simultaneously:

1. The airplane's velocity through the air (how fast it moves relative to the air)
2. The wind's velocity (how the air moves relative to the ground)

The airplane's actual path over the ground is the vector sum of these two velocities.

Compass Directions and Wind

In aviation, directions are often described using compass headings:

• North (N): 90° from the positive x-axis (straight up)
• East (E): 0° from the positive x-axis (right)
• South (S): 270° from the positive x-axis (down)
• West (W): 180° from the positive x-axis (left)

Example: Flying North with East Wind

Imagine a pilot wants to fly north to reach a city directly ahead. If there's a wind blowing from west to east (an easterly wind), the pilot faces an important decision:

• If the pilot points the airplane straight north (90°), the wind pushes the airplane eastward during flight
• The airplane ends up east of the intended destination

To compensate, the pilot must aim somewhat northwest (between 90° and 180°). This way:

• The wind pushes the airplane eastward
• The airplane's northwestward motion through the air carries it north and west
• These combine to create a straight northward path over the ground

Calculating Basic Angles

For simple cases, we can determine the general direction needed:

Example: An airplane flies at speed $v_{plane} = 100$ km/h through air. A wind blows east at $v_{wind} = 30$ km/h.