Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A plane is traveling 300 miles per hour. There is a 50 -mph wind. The angle between he velocity vector of the wind and the velocity vector of the plane is \(110^{\circ}\). (a) What is the component of the direction of the plane's motion? (b) In the absence of the wind but all else remaining the same, how fast would the plane be traveling?

Step-by-Step Solution

Answer

The component of the direction of the plane's motion is -172.12 mph. If there were no wind, the plane would be traveling at 311.8 mph.

Step 1: Find the component of direction of planes motion

The component of direction of the plane's motion that is in the direction of the plane's velocity is given by \(\text{Plane's speed} × \cos ( \text{angle between velocity vectors})\). This is \(300 \cos (110^{\circ})\). Applying the cosine of 110 degrees, we get -172.12 mph. The negative sign indicates that the wind hampers the plane's speed.

Step 2: Calculate velocity of the plane against the wind

The component of direction of the plane's motion that is perpendicular to the plane's velocity is given by \(\text{Plane's speed} × \sin ( \text{angle between velocity vectors})\). Thus, we get \(300 × \sin (110^{\circ})\), which equals 259.81 mph.

Step 3: Calculate plane's speed in absence of wind

The plane's speed in the absence of the wind can be calculated by using Pythagorean theorem: \(\sqrt{(\text{Speed against wind})^2 + (\text{Perpendicular speed})^2}\), substituting the above-obtained values, we get \(\sqrt{(-172.12)^{2} + (259.81)^{2}}\) = 311.8 mph.

Key Concepts

Velocity Vectors

Velocity vectors are fundamental in understanding the movement of objects like planes. A velocity vector shows the speed and direction of an object. For a plane, the velocity vector points in the direction the plane is moving and has a magnitude equal to its speed. In real-world scenarios, planes encounter additional elements like wind, impacting their velocity vector. To calculate the effective velocity of a plane heading into the wind, we break the velocity vectors into components.

• The plane's own velocity vector describes its speed and direction in still air.
• The wind's velocity vector adds or reduces speed depending on its interaction with the plane.

In the given problem, the wind forms a certain angle with the plane's velocity vector, allowing us to use trigonometric functions (cosine and sine) to decompose the velocity into components. Understanding these components helps us determine the plane's true motion under the influence of wind.

Plane Motion

Plane motion can be thought of as how an airplane moves through space, particularly when influenced by wind. Unlike simple straight-line motion, plane motion involves a vector that isn't static. It is important to separate the motions of the plane's engine power and wind influence.

• The component parallel to the direction of motion: This is where the plane's engine power directly aids movement, albeit in the presence of wind. Here, the cosine of the angle between the plane's movement and wind direction is utilized to find the effective forward speed.
• The component perpendicular to the direction of motion: This represents the influence of the wind's crosswind components on the plane's travel direction, calculated using the sine of the angle between the two vectors.

This separation helps better understand the effective trajectory and velocity of the plane, which can sometimes differ significantly from the plane's intended path.

Pythagorean Theorem

The Pythagorean theorem is a principle typically associated with right-angled triangles. In vector calculus, it applies to the relationship among vector components to find the overall magnitude. When considering the plane's complete velocity, even against factors like wind, we use this theorem to find the resultant speed.

• The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
• For our plane problem, the hypotenuse represents the plane’s velocity in the absence of wind.

By substituting the parallel and perpendicular components into this theorem, we calculate the plane's actual speed without the wind's interfering effects. This approach gives a clearer picture of pure plane speed, translating into real-world applications such as flight planning and safety.