Question

Airplane An airplane has an airspeed of 500 kilometers per hour \((\mathrm{km} / \mathrm{h})\) bearing \(\mathrm{N} 45^{\circ} \mathrm{E}\). The wind velocity is \(60 \mathrm{~km} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{W}\). Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?

Short answer

The groundspeed of the plane is approximately 494.2 km/h, and its direction is approximately 37.85° east of north.

Step-by-step solution

- Represent the Airplane's Velocity Vector

The airplane's airspeed is 500 km/h in the direction N 45° E. Convert this into component form. Using trigonometry, the north component is given by: . Therefore, . Similarly, the east component is given by: . Therefore, .

- Represent the Wind Velocity Vector

The wind velocity is 60 km/h in the direction N 30° W. Convert this into component form as well. The north component is: . Therefore, . The west component is: which points west, so it is negative in the east-west component: .

- Calculate the Resultant Vector

Add the components of both vectors to get the resultant vector. The north components add as and the east components add as .So, the resultant vector components are: .

- Find the Groundspeed

The groundspeed of the plane can now be found using the Pythagorean theorem for the resultant vector: .

- Determine the Direction

The direction of the plane relative to the ground is found using the arctangent function: .

Key concepts

Understanding Vectors

In mathematics and physics, vectors are quantities that have both magnitude (length) and direction. Examples include displacement, velocity, and force. Vectors can be represented graphically with an arrow or numerically using coordinates. For example, if a vector points north-east and has a magnitude of 500 km/h, we can break it down into north and east components using trigonometry. This helps us understand the vector's impact in different directions. In our problem, the airplane's airspeed and wind velocity are both vectors that need to be combined to find the plane's actual path.

Basic Concepts of Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is incredibly useful in this problem for converting between directional angles and vector components. Key trigonometric functions include sine (sin), cosine (cos), and tangent (tan). For example, to break down a vector into its horizontal and vertical components, you can use cosine for the horizontal part and sine for the vertical part. In the airplane problem, we use these functions to find the north and east components of both the airplane's airspeed and the wind's velocity.

What is a Resultant Vector?

A resultant vector is the vector produced as a result of adding two or more vectors together. It represents the cumulative effect of those vectors. When adding vectors, you add their respective components. Suppose you have two vectors: one representing the airplane's airspeed and the other representing the wind's velocity. Convert each vector into components and then add these components together. The resulting vector gives you the airplane's path relative to the ground, taking the wind into account. This results in both a new direction and magnitude, which tells us the groundspeed and the actual direction of travel.

Calculating Groundspeed

Groundspeed is the speed of an airplane relative to the ground, considering the effects of wind. To find it, first, determine the resultant vector from adding the airplane's airspeed vector and the wind velocity vector. Once the resultant vector is found, use the Pythagorean theorem to compute the magnitude of this vector, which gives you the groundspeed. For example, if your resultant vector components are in the form of \( R_x \) and \( R_y \), the groundspeed is calculated by \( \text{groundspeed} = \sqrt{R_x^2 + R_y^2} \). This value tells us how fast the airplane is moving relative to the ground, accounting for the impact of the wind.