Question

The pilot of an airplane wishes to fly due north, but there is a $36\mathrm{km}/\mathrm{h}$ wind blowing toward the west. (a) In what direction should the pilot head her plane if its speed relative to the air is $350\mathrm{km}/\mathrm{h}$ ? (b) Draw a vector diagram that illustrates your result in part (a). (c) If the pilot decreases the airspeed of the plane, but still wants to head due north, should she increase or decrease the angle found in part (a)?

Short answer

(a) 5.9° east of north; (b) Vector diagram with components; (c) Increase angle.

Step-by-step solution

Analyze the Problem

To solve part (a), we need the plane's velocity vector relative to the ground to be due north. The wind creates a velocity vector toward the west at 36 km/h. Hence, the plane's velocity vector relative to the air must have components to counteract the wind and maintain a northward direction.

Determine the Needed Velocity Components

The northward direction means the plane's ground velocity in the west direction should be zero: $$V_{pw,x}+V_w=0$$where $V_{pw,x}$ is the x-component of the plane's velocity relative to the air, and $V_w=-36\text{ km/h}$. Thus:$$V_{pw,x}=36\text{ km/h}$$

Apply Pythagorean Theorem

To solve for the angle $\theta$, use the plane's speed relative to the air. The speed forms the hypotenuse of a right triangle with a northward component (y-axis) needed to cancel the westward wind:$$V_{pw}=350\text{ km/h}$$Thus, using Pythagoras, the northward component is:$$V_{pw,y}=\sqrt{V_{pw}^2-V_{pw,x}^2}=\sqrt{{350}^2-{36}^2}$$

Calculate the Angle

Calculate $\theta$, the angle at which the plane should head, using the arctan function (where it shifts from due north):$$\theta ={\tan }^{-1}\left(\frac{V_{pw,x}}{V_{pw,y}}\right)$$

Solve the Equation for the Angle

Plug the values into the equation:$$V_{pw,y}=\sqrt{{350}^2-{36}^2}=\sqrt{122500-1296}=\sqrt{121204}\approx 348\text{ km/h}$$$$\theta ={\tan }^{-1}\left(\frac{36}{348}\right)\approx {\tan }^{-1}(0.1034)\approx {5.9}^{\circ }$$Thus, the pilot should head approximately ${5.9}^{\circ }$ east of north.

Create the Vector Diagram

Draw a diagram with: - A north-pointing vector representing the intended direction (ground velocity). - A westward vector for wind velocity. - An actual plane's velocity vector from the tail of the west vector pointing northeast, completing a right triangle.

Analyze Effect of Decreasing Airspeed (c)

If the pilot decreases the airspeed of the plane while trying to maintain a north direction, she must aim further east to overcome the same westward wind component. The angle $\theta$ should increase as less airspeed requires more compensatory angle to maintain a northward path.

Key concepts

Airplane Velocity

Airplane velocity is a crucial aspect of flight dynamics. It represents the speed and direction at which an airplane moves relative to the surrounding air. In our scenario, the airplane needs to achieve a specific velocity to reach a destination due north. This velocity is not just a straight line movement; it is a vector quantity having both magnitude (speed) and direction. Understanding velocity helps pilots control their aircraft accurately, aligning their flight path with navigation goals.
The initial desired vector position here is heading due north at a speed of 350 km/h. The challenge arises when external factors such as wind influence this path, necessitating adjustments for the pilot to achieve the intended northward direction.

Wind Speed Effect

Wind speed significantly affects an airplane's course. It has the power to drift the plane off its planned trajectory. In this exercise, the westward wind blowing at 36 km/h is an influencing factor. Pilots must adjust the airplane's heading to compensate for this wind effect to maintain a precise course. Without such adjustments, the wind would push the airplane westward, resulting in a deviation from its intended path.
To stay on course, the pilot must steer the aircraft slightly east of north. This compensatory adjustment balances the westward push from the wind, allowing the airplane to progress northward effectively despite the lateral wind force. Understanding wind's impact helps in planning accurate flight routes and ensuring safety in navigation.

Trigonometric Functions

Trigonometric functions are mathematical tools that help understand the relationships between angles and sides of triangles. In navigation and physics, these functions—such as sine, cosine, and tangent—are indispensable when dealing with vector quantities like airplane velocity. They enable us to decompose complex movements into understandable components.
For this problem, the tangent function is specifically used to resolve the angle at which the aircraft should head to counteract wind. By calculating the arctan of the velocity components, the angle $\theta$ is determined, showcasing how trigonometric relationships underlie critical navigational adjustments. Mastery of these functions allows pilots and engineers to solve complex flight vector problems efficiently.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry that relates the lengths of sides in a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this exercise, the theorem is applied to determine the northward velocity component of the airplane.
Given the airplane's speed forms the hypotenuse of the triangle, with one side being the westward wind component, the theorem helps find the plane's northward velocity. The calculation $$V_{pw,y}=\sqrt{V_{pw}^2-V_{pw,x}^2}$$ gives us the northward component necessary for the equation. This method is key to resolving real-world navigation problems where precise calculations are critical.

Directional Angles

Directional angles in navigation determine the direction in which an aircraft needs to be steered. They are measured relative to the north, helping pilots make precise course adjustments. In this problem, the angle $\theta$ is calculated to offset the westward wind, allowing the plane to move correctly due north.
The angle $\theta$ is determined using the formula $$\theta ={\tan }^{-1}\left(\frac{V_{pw,x}}{V_{pw,y}}\right)$$. By computing this angle, pilots know exactly how much to adjust their heading. As conditions change, such as a decrease in airspeed, this angle must be recalculated to maintain the desired flight direction. Understanding directional angles is essential for accurate navigation and ensuring the safety of the flight path.