Vectors in 2D: Problem Solving

Wind combines with the plane's intended velocity to create ground velocity, the actual path traveled. A plane flying north at 180 km/h with a crosswind will deviate from its intended course. The ground velocity results from vector addition of the plane's airspeed and wind velocity, forming a vector triangle. This combined motion determines both the plane's actual speed and direction relative to the ground.

Ground velocity equals the plane's intended velocity plus wind velocity. When a plane travels 80 kilometers in 30 minutes while aiming north, this displacement reveals the combined effect of both velocities. By subtracting the intended velocity from the ground velocity, you isolate the wind's contribution. This vector subtraction reveals wind magnitude and direction, essential for understanding navigation errors.

Wind velocity is found by resolving ground velocity into components and subtracting the intended velocity. The northward and eastward components of ground velocity are separated, then the northward intended velocity is subtracted from the northward component. The remaining northward and eastward components form a right triangle, where the wind velocity magnitude equals the hypotenuse calculated using the Pythagorean Theorem.

The wind pushes the plane eastward, so the pilot must aim west of north to counteract this drift. By introducing a westward velocity component, the plane's adjusted heading cancels the wind's eastward influence. A 4.4-degree westward adjustment creates a resultant velocity pointing due north, ensuring the plane follows its intended path despite wind interference.

Resolving velocity into northward and eastward components isolates wind's directional effect. Ground velocity components show the plane traveled mostly north but also eastward, revealing wind's eastward push. By comparing intended and actual velocity components, the wind's magnitude and direction become clear. This component analysis demonstrates how wind introduces both speed reduction and directional deviation.

The plane's ground displacement results from adding its airspeed vector to the wind velocity vector. Flying north at 180 km/h for 30 minutes would cover 90 kilometers, but wind deflects the path eastward, reducing ground speed to 160 km/h and creating an 80-kilometer displacement. This vector sum demonstrates how two velocity vectors combine to produce the actual path traveled.

The wind velocity magnitude is approximately 24.9 kilometers per hour, acting consistently throughout the 30-minute flight. This wind speed was calculated by analyzing the difference between the plane's intended northward path and its actual ground displacement. The wind's eastward component caused the 5-degree deviation, demonstrating how relatively modest wind speeds significantly alter aircraft trajectories.