Question

An aircraft has to fly between two cities, one of which is \(600.0 \mathrm{km}\) north of the other. The pilot starts from the southern city and encounters a steady \(100.0 \mathrm{km} / \mathrm{h}\) wind that blows from the northeast. The plane has a cruising speed of \(300.0 \mathrm{km} / \mathrm{h}\) in still air. (a) In what direction (relative to east) must the pilot head her plane? (b) How long does the flight take?

Short answer

(a) The plane must head 13.7° north of east. (b) The flight takes approximately 8.5 hours.

Step-by-step solution

Understand the Problem

The plane must travel northwards with a 600 km distance. There's also a wind blowing 100 km/h from the northeast, which is at a 45-degree angle towards the south-west direction.

Visualize the Vectors

Imagine the north as the y-axis and east as the x-axis. The plane's velocity vector in still air is represented as \((V_{plane}, 0) = (300 \ \mathrm{km/h}, 0)\) when heading east. The wind velocity vector from the northeast is \((V_{wind}, V_{wind}) = (100 \cos 45^\circ, -100 \sin 45^\circ)\).

Resolve the Wind Component

Find the components of the wind velocity: \(V_{wind-x} = 100 \cos 45^\circ = 70.7 \ \mathrm{km/h}\), and \(V_{wind-y} = -100 \sin 45^\circ = -70.7 \ \mathrm{km/h}\). This means the wind is pushing the plane southwest.

Equate Resultant Velocities

Since the plane must head north, the resultant velocity in the y-direction must be equal to the cruising speed in still air plus the wind y-component. Solve \(V_{plane,y} + V_{wind,y} = 0\) leading to \(V_{plane,y} = 70.7 \ \mathrm{km/h}\). Find the x-component of the plane's velocity using \V_{plane}^2 = V_{plane,x}^2 + V_{plane,y}^2\ implying \(300^2 = V_{plane,x}^2 + 70.7^2\).

Calculate Required Heading Direction

Solve for \(V_{plane,x}\): you get approximately \(292.2 \ \mathrm{km/h}\). Use the tangent of the angle to find the direction: \ an^{-1}\left(\frac{70.7}{292.2}\right) \approx 13.7^\circ\. Therefore, the pilot needs to head approximately 13.7° north of east.

Calculate the Effective Speed Towards Target

The effective northward (y-direction) speed of the plane is unaffected by the x-component of the wind. Hence, the effective speed in the northward direction is 70.7 km/h.

Determine Flight Time

Use the formula time = distance/speed to find the time: \(t = \frac{600}{70.7} \approx 8.5 \ \mathrm{hours}\). Thus, the flight will take approximately 8.5 hours.

Key concepts

Vector Components

Understanding vector components is crucial for airplane navigation. Imagine you are navigating a plane, and you need to account for different forces acting on it. In simple terms, vectors are arrows that show both direction and magnitude. Here, you have two main vectors to consider: the plane's velocity and the wind's velocity.

The plane's velocity vector is straightforward when it is cruising in still air — it simply follows the path it is facing, like a line pointing straight ahead. In this task, we imagine the plane traveling north as a vertical vector on a graph. Now, the wind complicates things. The wind blows from a specific angle, here northeast to southwest, impacting both the north-south direction and the east-west direction.

When resolving vectors, especially in a two-dimensional plane, break them down into components along the x-axis (east-west) and y-axis (north-south). This means you can measure each vector's effect on moving north, south, east, or west, then sum up these effects to see the overall direction and speed.

Wind Effect on Flight

Wind plays a significant role in determining a plane's path. In this exercise, wind is blowing from the northeast at 100 km/h. The trickiest part is that it does not push the plane straight backward or forward. Instead, it pushes in a 45-degree angle towards the southwest, influencing both the northward and eastward movement of the plane.

To handle this, calculate the wind's components: how much it affects horizontally (across the x-axis) and vertically (along the y-axis). For a 45-degree wind, you'll find the effect to be equal in both directions. Here, the wind affects each direction by about 70.7 km/h. Thus, this wind component pushes the plane a bit off course, shifting it southward and impacting the effective speed towards its northern target.

In response, the pilot must adjust the heading direction slightly north of east to counteract the wind's southwestern push. The entire task of piloting through wind involves constantly balancing these forces to maintain the desired course.

Flight Time Calculation

Once you have resolved vector directions and understood wind effects, it is time to calculate how long the journey will take. The key here is to determine the effective speed in the direction you want to travel — which, in this case, is north.

The problem requires computing the effective northward speed after accounting for the wind. Though the wind changes direction by pushing the plane slightly southwest, the plane’s own engine thrust can keep part of the motion northward unaffected.

Here, through vector calculations, we find the effective northward speed to be 70.7 km/h. To compute the time, use the formula:

• Time = Distance / Speed

For a 600 km journey at a northward speed of 70.7 km/h, the flight time turns out to be about 8.5 hours. Hence, knowing the right speed and distance helps plan fuel, resources, and timings accurately.