A plane is flying at a speed of 320 miles per hour on a bearing of N70 ^∘ E. Its ground speed is

A plane is flying at a speed of 320 miles per hour on a...

A plane is flying at a speed of 320 miles per hour on a bearing of N70 $^{\circ} \mathrm{E}$. Its ground speed is 370 miles per hour and its true course is $30^{\circ} .$ Find the speed, to the nearest mile per hour, and the direction angle, to the nearest tenth of a degree, of the wind.

Key Concepts

Wind Triangle

The wind triangle is a navigational tool used to solve for unknown quantities such as wind speed and direction by forming a triangle from the aircraft’s airspeed vector, the wind vector, and the ground speed vector. By applying concepts from trigonometry and vector decomposition, the wind triangle provides a systematic method to determine how wind affects the flight path, enabling pilots and navigators to adjust their heading accordingly.

Vector Addition

Vector addition involves combining two or more vectors, such as velocity vectors of an aircraft and the wind, to obtain a resultant vector. In navigation problems, the resultant vector gives the actual ground speed and track of the aircraft, and understanding vector addition is essential for determining how external influences like wind affect a vehicle’s motion.

Decomposition of Vectors

Decomposing vectors into their components involves breaking down a vector into its horizontal and vertical parts, usually along the east-west and north-south axes. This process is crucial in navigation as it allows one to calculate the effects of wind and airspeed in different directions, and to properly add or subtract these effects to determine the resultant ground vector.

Bearings and Navigational Angles

Bearings are a way of describing direction relative to the cardinal points, such as north or east, and are used in navigation to indicate the course of a vehicle. Understanding bearings and how to convert them into standard angle measures or vector components is vital, as it ensures the correct representation of the direction for both the aircraft’s heading and the wind’s influence.

Transcript

00:01 In this question, it is given that a plane is flying at a speed of 320 miles per hour on a bearing of an 70 degree e.

00:11 Its ground speed is 370 miles per hour and its true course is 30 degree.

00:18 We need to find the speed to the nearest mile per hour and the direction angle to the nearest tenth of a degree of the wind.

00:29 So let's see how to solve this question.

00:30 So, based on the available information, we can draw the vector diagram for the plane speed and the wind speed as shown below.

00:42 So this is a vector diagram.

00:44 Here, v represents the plane speed, w represents the wind speed and v plus w represents the resultant speed.

00:56 Now for the vector v, the direction angle theta from the horizontal can be calculated.

01:04 As theta is equal to 90 degree minus 70 degree.

01:13 Hence this will be equal to 20 degree.

01:18 We know that if we have the magnitude and direction angle then a non -zero vector can be represented as v is equal to magnitude v.

01:30 Cosine theta i plus magnitude v sine theta j.

01:40 Therefore, the velocity of the plane v can be written as 320 miles per hour, which is given cosine 20 degree i plus 320 sine 20 degree j.

02:04 And when we further calculate this, we get velocity of the plane vector v is equals to 300 .70i plus 109 .45j.

02:26 Since it is given that the ground speed is 370 miles per hour and its true course is 30 degree.

02:35 Therefore, the vector representation of v plus w can be written as 370 miles per hour, which is given, cosine, this angle is 30 degree, 30 degree i plus 30070 sign 30 degree j.

03:06 When you further calculate this, we get v plus w is equals to 320 .43i plus 185j.

03:25 This was equation 1 and this one is equation 2.

03:37 And now let's write the vector representation of velocity of wind...

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