Finding the Actual Speed and Direction of an Aircraft A...

Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains a constant airspeed of 550 miles per hour $(\mathrm{mph})$ headed due north. The jet stream is $100 \mathrm{mph}$ in the northeasterly direction. (a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the 747 relative to the air and the velocity $\mathbf{v}_{\mathrm{w}}$ of the jet stream in terms of $\mathbf{i}$ and $\mathbf{j}$. (b) Find the velocity of the 747 relative to the ground. (c) Find the actual speed and direction of the 747 relative to the ground.

Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains a constant airspeed of 550 miles per hour $(\mathrm{mph})$ headed due north. The jet stream is $100 \mathrm{mph}$ in the northeasterly direction.
(a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the 747 relative to the air and the velocity $\mathbf{v}_{\mathrm{w}}$ of the jet stream in terms of $\mathbf{i}$ and $\mathbf{j}$.
(b) Find the velocity of the 747 relative to the ground.
(c) Find the actual speed and direction of the 747 relative to the ground.

Key Concepts

Vector Representation with Unit Vectors

This concept involves expressing any vector, such as velocity, in terms of its orthogonal components along specified directions, typically represented by unit vectors i and j. In a Cartesian coordinate system, i commonly denotes the unit vector in the east direction and j denotes the unit vector in the north direction. Understanding this allows one to break down a given velocity into its constituent parts, facilitating mathematical operations like addition and subtraction of vectors.

Relative Velocity

Relative velocity is the idea that the observed velocity of an object depends on the frame of reference in which it is measured. In problems involving an aircraft and wind, the actual velocity relative to the ground is obtained by vectorially adding the aircraft's velocity relative to the air and the wind's velocity. This concept is essential to accurately determine motion characteristics in dynamics.

Vector Addition and Resultant Calculation

Vector addition is the process of combining two or more vectors to determine the resultant vector. In the context of velocity, it involves summing the corresponding i and j components of each vector. Once the components of the resultant are known, one can compute its magnitude using the Pythagorean theorem and determine its direction using inverse trigonometric functions, typically arctan, which gives the angle relative to one of the axes.

Transcript

00:01 Plane flying north at 550 miles per hour, and then the wind or the jet stream is blowing northeast at 100 miles per hour.

00:08 We want to find the velocity vector for the plane, the wind, the ground speed, and then the actual speed and direction of the plane.

00:16 Okay, so first of all, the north vector is 0 .i plus 1j.

00:29 So the vector of the plane is 550 times that, 0i plus 1j or 550.

00:48 All right.

00:49 Now, when they say the wind is blowing northeast, they mean exactly north and east, halfway between.

00:59 So at a 45 degree angle, that's northeast.

01:05 And so that triangle is 1 -1 square root of 2.

01:11 So northeast then is 1 over the square root of 2 i plus 1 over the square root of 2j.

01:22 So the vector for the wind is 100.

01:26 I'm going to convert that to its nicer form, square root of 2 over 2i plus square to 2 over 2j, which gives me 50 i, no 50 square root of 2i plus 50 square root of 2 j okay so then the speed relative to the ground is the sum of those two vectors so it's 50 square to 2 i plus 550 plus 50 square to 2 j all right, so to find the actual speed now, it's the magnitude of the velocity of the ground speed one.

02:30 So it's the square root of 50 square root of 2 squared plus 550 plus 50 square root of 2 squared.

02:44 So that's 5 ,000 plus 550 squared, this number, 3 of 2.

03:02 02500 plus 550 times 50 times 2 55000 square root of 2 plus 5 ,000 so 312000 plus 5500 squared a 2.

03:34 Okay, so that's the exact answer, but that's not very useful.

03:39 So let's convert that to a decimal square root of two times 5500 ,000 i mean, plus 3125000.

03:52 No, 2 mean zeros...

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