An airplane pilot checks her instruments and finds that the air speed of the plane is 325 km/h. the instruments show that the plane is pointed in a

direction (S 30 W). A radio report indicates that the wind velocity is 80 km/h (W). What is the velocity of the plane relative to the ground as it would be recorded by an air traffic controller in a nearby airport?

Well, it sounds like the pilot is on the airplane, not flying it. I hope she's not planning on directing the plane with her remote control! Anyways, let's calculate the velocity of the plane relative to the ground.

To do that, we need to break down the velocity into its components. The air speed of 325 km/h is the speed of the plane relative to the air around it. The wind velocity of 80 km/h blowing from the west is the speed of the wind relative to the ground.

Now, we can use some good old trigonometry to solve this problem. Since the plane is pointed in the direction S 30 W, we can decompose the air speed into its southward and westward components.

The southward component can be found using the cosine function, which gives us cos(30°) = 0.866. So the southward component of the air speed is 325 km/h * 0.866 = 281.45 km/h.

The westward component can be found using the sine function, which gives us sin(30°) = 0.5. So the westward component of the air speed is 325 km/h * 0.5 = 162.5 km/h.

Now, to find the velocity of the plane relative to the ground, we just need to add the wind velocity to the components of the air speed. Since the wind is blowing from the west with a velocity of 80 km/h, the westward component of the plane's velocity relative to the ground will be 162.5 km/h + 80 km/h = 242.5 km/h.

So, according to my calculations, the velocity of the plane relative to the ground, as it would be recorded by an air traffic controller, is approximately 242.5 km/h. Now that's some fast and funny aviation math!

To solve this problem, we need to use vector addition.

Let's break down the velocities and directions given:

1. Air speed of the plane: 325 km/h
- This is the speed at which the plane is moving through the air.

2. Direction of the plane: S 30 W
- This means the plane is pointed in the direction of 30 degrees west of south.

3. Wind velocity: 80 km/h (W)
- The wind is blowing from the west.

To find the velocity of the plane relative to the ground, we need to add the velocities of the plane and the wind together.

Step 1: Convert the directions to components:
- East (E) is positive, West (W) is negative
- North (N) is positive, South (S) is negative

The direction S 30 W can be broken down as:
- South: -30 degrees from the positive x-axis
- West: -90 degrees from the positive x-axis

Step 2: Calculate the x and y components of the air speed:
- Note that the air speed is always in the direction the plane is pointing.
- Since the direction is southwest, it can be broken down into the x and y components using trigonometry.
- The horizontal component (x) = air speed * cos(angle)
- The vertical component (y) = air speed * sin(angle)

In this case:
- x = 325 km/h * cos(-30 degrees)
- y = 325 km/h * sin(-30 degrees)

Step 3: Calculate the x and y components of the wind velocity:
- The wind is blowing from the west, so it only has an effect in the x-direction.
- The wind's velocity will be negative since it is blowing in the opposite direction of the plane's motion.
- The horizontal component (x) of the wind = -80 km/h
- The vertical component (y) of the wind = 0 km/h (since wind doesn't affect the vertical motion of the plane)

Step 4: Add the x and y components of the air speed and wind velocity to get the velocity of the plane relative to the ground:
- x_component_plane = x_component_air_speed + x_component_wind
- y_component_plane = y_component_air_speed + y_component_wind

In this case:
- x_component_plane = x + x_wind
- y_component_plane = y + y_wind

Step 5: Calculate the magnitude and direction of the velocity of the plane relative to the ground:
- magnitude_plane = sqrt(x_component_plane^2 + y_component_plane^2)
- direction_plane = arctan(y_component_plane / x_component_plane)

In this case:
- magnitude_plane = sqrt(x_component_plane^2 + y_component_plane^2)
- direction_plane = arctan(y_component_plane / x_component_plane)

Therefore, to find the velocity of the plane relative to the ground as recorded by an air traffic controller, we need to perform these calculations.

To find the velocity of the plane relative to the ground, we need to consider the effect of wind on the plane's motion. We can break down the problem into two components: the plane's velocity relative to the air and the wind's velocity.

1. First, let's find the plane's velocity relative to the air. We are given that the airspeed (velocity relative to the air) is 325 km/h, and the direction the plane is pointed in is S 30 W.

To make calculations easier, let's convert the compass direction to mathematical angles. South corresponds to 180 degrees, and West corresponds to 270 degrees. Therefore, S 30 W can be converted to 180 + 30 = 210 degrees.

To find the components of the velocity relative to the air, we'll use trigonometry. The east-west component can be found by multiplying the airspeed by the cosine of the angle, and the north-south component can be found by multiplying the airspeed by the sine of the angle. So:

East-West component = 325 km/h * cos(210 degrees)
North-South component = 325 km/h * sin(210 degrees)

2. Next, let's consider the wind velocity. We are given that the wind velocity is 80 km/h to the west. Again, let's convert this into components: 80 km/h * cos(270 degrees) for the east-west component and 80 km/h * sin(270 degrees) for the north-south component.

3. Finally, we can find the velocity of the plane relative to the ground using vector addition. The east-west component of the velocity relative to the ground is the sum of the east-west components of the velocity relative to the air and the wind velocity, while the north-south component is the sum of the north-south components.

Velocity relative to the ground (east-west) = East-West component (velocity relative to the air) + East-West component (wind velocity)
Velocity relative to the ground (north-south) = North-South component (velocity relative to the air) + North-South component (wind velocity)

Simply calculate these two components separately to find the final velocity of the plane relative to the ground.

Vp + 80 = 325km/h[240o] CCW.

Vp + 80 = -162.5-281.5i,
Vp = -242.5 - 281.5i = 372km/h[49.3o] S. of W.