An aircraft is to fly to a destination \(800 \mathrm{~km}\) due north of its starting point. Its airspeed is \(800 \mathrm{~km} \mathrm{~h}^{-1}\). The wind is from the east at a speed of \(30 \mathrm{~m} \mathrm{~s}^{-1}\). On what compass heading should the pilot fly? How long will the flight take? If the wind speed increases to \(50 \mathrm{~ms}^{-1}\), and the wind backs to the north-east, but no allowance is made for this change, how far from its destination will the aircraft be at its expected arrival time, and in what direction?

Understanding the effects of wind on aircraft navigation is crucial for safe and accurate flight paths. When assessing aircraft movement, airspeed and wind speed vector analysis plays a pivotal role. Imagine an aircraft trying to fly to a specific point; its airspeed represents its velocity in still air, whereas the wind adds another layer of motion affecting its actual path over the ground.

When a plane encounters wind from any direction, the wind speed must be treated as a vector, just like the aircraft's airspeed. A vector is a quantity having both magnitude and direction. By combining these vectors, we can determine their resultant vector, which illustrates the actual velocity and trajectory of the aircraft relative to the ground.

In our exercise, the pilot must account for a wind coming from the east at 108 km/h after converting the given wind speed into the same units as the airspeed. Airspeed is a vector pointing to the north with a magnitude of 800 km/h, and wind speed becomes a vector pointing to the west. To correctly navigate, the pilot will adjust the aircraft's heading to counteract the wind's influence.

Calculating an aircraft's flight trajectory is essential for determining the correct heading required to reach a predetermined destination. It involves taking into account the aircraft's airspeed vector alongside any wind speed vectors and finding their resultant to predict the actual course taken over the ground.

In flight trajectory calculation, we often visualize vectors as arrows pointing in the direction of motion. The length of these arrows represents the speed's magnitude. To find the actual course (the trajectory), one needs to draw these vectors, typically using a scale diagram or by employing mathematical methods like vector addition. In our example, the pilot originally intends to fly due north, and the east winds push the aircraft towards the west. By plotting these vectors and considering their combined effect, the pilot can adjust the aircraft's direction to compensate for wind drift and stay on the desired northerly course.

This adjustment results in a new heading, which is crucial for ensuring that the aircraft arrives at its destination, and not miles off course. Understanding and calculating this proper heading, influenced by both airspeed and wind, is what defines the pilot's flight trajectory.

The resultant vector magnitude and direction are the key outcomes of any vector analysis in aircraft navigation. The magnitude represents the combined speed of the aircraft considering both its airspeed and the wind speed, while the direction indicates the actual path the aircraft should maintain to reach its target point.

In the provided solution, we applied the Pythagorean theorem to compute the resultant vector magnitude, taking into consideration the right-angled triangle formed by the northward airspeed vector and the eastward wind speed vector. The magnitude tells us how fast the aircraft is effectively moving across the ground. In our case, the magnitude came out to be approximately 812 km/h. The pilot needs to maintain this speed relative to the ground to reach the destination on time.

The resultant vector's direction, derived from trigonometry using the arctangent function, gives the angle at which the pilot must fly to counteract the wind effect, which was approximately 7.7 degrees west of north. Understanding these concepts is vital for pilots as they allow for accurate and timely arrivals at their destinations, as well as ensuring the overall safety of the flight.