11.9
Vectors in 2D: Problem Solving
A plane traveling due north at 180 km/h in still air was found to be 80 km off-course after 30 minutes, deviating approximately 5 degrees east of nort…
A plane is flying due north at a constant airspeed of 180 kilometers per hour. After 30 minutes, it is found 80 kilometers away, but 5 degrees east of north. This means the wind has changed the plane’s intended path.
The goal is to find the wind’s velocity and the direction the plane should fly to stay on course.
The plane’s ground velocity comes from combining its intended velocity with the wind’s velocity, forming a vector triangle.
The plane covers 80 kilometers in 30 minutes, giving the magnitude of its ground velocity. Since this velocity points 5 degrees east of north, it can be resolved into two components: northward and eastward.
The intended velocity is due north at 180 kilometers per hour. Subtracting this from the ground velocity gives the wind velocity. Its magnitude is found using the Pythagorean Theorem.
Finally, to stay on course, the pilot must aim slightly west of north to cancel the wind’s effect.
This adjustment is found by resolving the adjusted velocity into components and matching the westward part to the wind’s eastward push. The result shows the plane must aim about 4.4 degrees west of north to stay on course.
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Q1: How does wind affect a plane's ground velocity and actual path?
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.
Q2: What is the relationship between displacement, ground velocity, and wind velocity?
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.
Q3: How do you calculate wind velocity magnitude using the Pythagorean Theorem?
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.
Q4: Why must a pilot adjust heading west of north to maintain a northward course?
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.
Q5: What does resolving velocity into components reveal about wind direction?
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.
Q6: How does vector addition explain the plane's 80-kilometer displacement in 30 minutes?
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.
Q7: What is the estimated magnitude of the wind velocity in this navigation problem?
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.