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

Finding the Actual Speed and Direction of an Aircraft An Airbus A320 jet maintains a constant airspeed of $500 \mathrm{mph}$ headed due west. The jet stream is $100 \mathrm{mph}$ in the southeasterly direction. (a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the A320 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 A320 relative to the ground. (c) Find the actual speed and direction of the A 320 relative to the ground.

Finding the Actual Speed and Direction of an Aircraft An Airbus A320 jet maintains a constant airspeed of $500 \mathrm{mph}$ headed due west. The jet stream is $100 \mathrm{mph}$ in the southeasterly direction.
(a) Express the velocity $\mathbf{v}_{\mathrm{a}}$ of the A320 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 A320 relative to the ground.
(c) Find the actual speed and direction of the A 320 relative to the ground.

Key Concepts

Vector Addition

Vector addition is the process of combining multiple vectors by adding their corresponding components. This is essential when calculating the net effect of different velocity vectors, as it provides the required resultant vector which represents the overall motion.

Magnitude and Direction Calculation

After vector addition, the overall speed (magnitude) of the resultant vector is determined using the Pythagorean theorem, and its direction is found by calculating the angle with trigonometric functions such as the inverse tangent, helping to translate component information back into a comprehensible speed and direction format.

Relative Velocity

Relative velocity is the idea of determining an object's velocity by considering the contribution of multiple velocity vectors from different reference frames. In problems like this, the aircraft's velocity relative to the ground is found by vectorially adding its velocity relative to the air and the wind’s velocity.

Vectors and Component Representation

This concept involves expressing physical quantities such as velocity in terms of components along chosen coordinate axes, typically using i (east) and j (north). It allows you to break down a vector into horizontal and vertical parts, making calculations involving direction and magnitude straightforward.

Transcript

00:01 So you have a jet that is flying at 500 miles per hour going due west.

00:07 So its vector, we'll call that the jet vector, would be equivalent.

00:15 Well, let's use a capital j so it doesn't look like that ij notation.

00:22 And we would have that being negative 500i, and that's plus 0j.

00:30 And the jet stream is traveling southeast at only 100.

00:37 So let's draw that in red.

00:39 And it's doing this.

00:40 So the air vector is going to end up being, we know that this is a magnitude of 100.

00:51 And so this is, this leg of this is going to end up being 100 divided by the square root of 2.

00:58 And this leg is going to have a magnitude of 100 divided by the square root of 2.

01:03 And i'm just going to leave it in that notation right now and not going to worry about simplifying it out.

01:08 I'm just going to put it like that, and you can fix that if need b.

01:11 And that's going to be positive i, and that's going to be a negative j.

01:23 And now on part b, we want to find what the vector is, the actual ground vector for this, this point.

01:33 Plane and that vector is going to be this vector.

01:38 And so we want to figure out by adding these components together.

01:42 And so now i'll add the components to give what that new velocity vector is for that plane.

01:48 So we're going to take that negative 500 and we'll add to that that 100 divided by the square root of 2.

01:58 And that comes out to be negative 429 .29 .29 .9...

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