An airplane's velocity with respect to the air is 580 miles per hour, and it is heading N 60 degrees W. The wind, at the altitude of the plane, is from the southwest and has a velocity of 60 miles per hour. What is the true direction of the plane, and what is its speed woth respect to the ground?

I have no idea where to start!

North speed = v

East speed = u

Wind u = 60 cos 45 = 42.4
Wind v = 60 sin 45 = 42.4

plane u = -580 sin 60 = -502
plane v = +580 cos 60 = +290

total u = 42.4 - 502
total v = 42.4 + 290

now do sqrt (u^2 + v^2)
and tan^-1 (v/u)

An airplane's velocity with respect to the air is 580 mph and it is headed towards N 58° W. The wind, at the altitude of the plane, is from the southwest and has a velocity of 60 mph. What are the resultant speed and direction of the airplane?

Thank you!!

I need the answer

Well, it sounds like this airplane is having quite the dance party with the wind! Let's break it down step by step, like a clumsy clown trying to do a pirouette.

First, we need to find the airplane's true direction. N 60 degrees W means it's heading 60 degrees west of north. So, if you imagine the north as the starting point on a compass, you would turn 60 degrees towards the west. It's like giving the plane some swag in its step.

Now, let's consider the wind. The wind is coming from the southwest, which is 45 degrees south of west. So, it's like the wind is saying, "Hey plane, I'm blowing at you from an angle!"

Since we know both the airplane's heading (60 degrees west of north) and the wind's direction (45 degrees south of west), we can figure out the angle between the airplane's heading and the wind. We just need to do some math magic: 60 degrees + 45 degrees = 105 degrees.

Hold on tight, because we're about to do some trigonometry acrobatics. To find the true direction of the plane, we need to subtract the wind angle from the plane's heading. So, 60 degrees - 105 degrees equals -45 degrees.

But wait, what does -45 degrees mean? It basically means the plane is heading 45 degrees to the right of south. So, the true direction of the plane is S 45 degrees W.

Now, let's move on to finding the speed of the plane with respect to the ground. To do that, we need to consider the effect of the wind on the plane's velocity.

The plane's velocity with respect to the air is 580 miles per hour, and the wind is blowing at 60 miles per hour. Think of it as the wind giving the plane a little boost or hindrance, depending on the direction.

To find the speed of the plane with respect to the ground, we need to combine the plane's velocity and the wind's velocity. Since the wind is blowing from the southwest (45 degrees south of west) and the plane is heading N 60 degrees W (45 degrees north of west), we have two vectors we can add together.

Using some trigonometry tricks again, we can find the magnitude of the resulting vector (speed) by using the Pythagorean theorem. Consider the plane's velocity as the x-component and the wind's velocity as the y-component.

So, the speed of the plane with respect to the ground is the square root of (580^2 + 60^2), which is approximately 582.27 miles per hour.

So, there you have it! The true direction of the plane is S 45 degrees W, and its speed with respect to the ground is about 582.27 miles per hour. Bon voyage! Or should I say, "Bon flight"?

To find the true direction of the plane and its speed with respect to the ground, we can break down the velocities and use vector addition.

Step 1: Draw a diagram illustrating the given information. Label the direction of the plane's velocity with respect to the air as N 60° W, and the direction of the wind as coming from the southwest.

Step 2: Convert the given directions into a standard compass direction. Since N 60° W is in the northwest quadrant, it can be converted to W 30° N.

Step 3: Convert all velocities to vector form. The plane's velocity with respect to the air is 580 mph in the direction of W 30° N, and the wind velocity is 60 mph in the direction from the southwest (S 45° W).

Step 4: Perform vector addition. Add the vectors representing the plane's velocity with respect to the air and the wind velocity.

To add vectors, you can use trigonometry. You can decompose each vector into its x (West-East) and y (South-North) components. Then add the x-components and the y-components separately.

For the plane's velocity:
- The x-component = 580 mph * cos(30°) = 580 * √3/2
- The y-component = 580 mph * sin(30°) = 580 * 1/2

For the wind velocity:
- The x-component = 60 mph * cos(45°) = 60 * √2/2
- The y-component = 60 mph * sin(45°) = 60 * √2/2

Add the x-components together and the y-components together.

The x-component of the resulting vector is (580 * √3/2) + (60 * √2/2).
The y-component of the resulting vector is (580 * 1/2) - (60 * √2/2).

Step 5: Determine the magnitude and direction of the resulting vector. Use the formula:
Magnitude = √(x-component^2 + y-component^2)
Direction = arctan(y-component / x-component)

Plug in the values and calculate the magnitude and direction of the resulting vector.

Once you have the magnitude and direction of the resulting vector, you will have the true direction of the plane and its speed with respect to the ground.