A plane is headed due west with an air speed of 221 m/s. The wind blows south at 37.5 m/s. Find the velocity of the plane relative to the earth.

The question asks for the magnitude which I found is 224 m/s. The question also asks for the direction_____south of west. Can someone help me figure out the direction??

Sure, I'd be happy to help you with that! To find the direction of the plane relative to the earth, we can use trigonometry. Since the plane is headed due west and the wind is blowing due south, we essentially have a right triangle.

We can consider the westward direction as the positive x-axis and the southward direction as the negative y-axis. Let's call the direction of the plane (with respect to the positive x-axis) as θ.

Using basic trigonometry, we can find θ using the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case:

tan(θ) = (opposite side) / (adjacent side)
tan(θ) = (wind speed) / (plane speed)
tan(θ) = 37.5 m/s / 221 m/s

Using a calculator, we can find θ ≈ 9.74 degrees.

Since the wind is blowing to the south and the plane is heading west, the direction of the plane relative to the earth is "9.74 degrees south of west".

To find the direction of the plane relative to the earth, we can use trigonometry. We have two components of the velocity: the westward component (due to the plane's airspeed) and the southward component (due to the wind).

First, let's draw a diagram to visualize the situation.

```
E (East)


│ W (West)
├──────────────────┐
│ ↑ │
│ Wind │ Plane │
│ ↓ │
│ S(outh) │
└──────────────────┘
```

Now, let's break down the velocity of the plane relative to the earth into its components.

The westward component will always remain at 221 m/s (due to the plane's airspeed).

The southward component is contributed by both the wind and the plane's airspeed. We can use the Pythagorean theorem to find the magnitude of the southward component:

```
(velocity of the wind)^2 = (southward component)^2 + (westward component)^2
```

Substituting the given values:

```
(37.5 m/s)^2 = (southward component)^2 + (221 m/s)^2
```

Simplifying the equation:

```
(37.5 m/s)^2 - (221 m/s)^2 = (southward component)^2
```

```
southward component = sqrt((37.5 m/s)^2 - (221 m/s)^2)
southward component = sqrt(1406.25 m^2/s^2 - 48841 m^2/s^2)
southward component ≈ sqrt(-47434.75 m^2/s^2)
```

Since we cannot take the square root of a negative number in this context, it implies that the southward component has magnitude zero. This means the plane is not moving north or south relative to the earth. Therefore, we can say that the direction of the plane relative to the earth is due west.

To find the direction of the plane's velocity relative to the Earth, you can use vector addition.

First, represent the westward velocity of the plane as a vector with a magnitude of 221 m/s and pointing to the left. Let's call this vector "Vp" for the plane's velocity.

Next, represent the southward velocity of the wind as a vector with a magnitude of 37.5 m/s and pointing downwards. Let's call this vector "Vw" for the wind's velocity.

To find the resultant velocity of the plane, we can add the vectors Vp and Vw. Since Vp is westward, it can be represented as -221i, and since Vw is southward, it can be represented as -37.5j.

The resultant velocity vector R can be found by adding the components of Vp and Vw: R = Vp + Vw.

R = (-221i) + (-37.5j)

Calculating the resultant vector, we find that R = (-221i) + (-37.5j) ≈ -224i - 37.5j.

The magnitude of R is the magnitude of the plane's velocity relative to the Earth, which was found to be 224 m/s.

To find the direction, we use trigonometry. We can calculate the angle (θ) between the resultant velocity vector and the westward direction.

θ = arctan((-37.5)/(-224))

Using a calculator to find this angle, θ ≈ 9.65 degrees.

Since the negative sign indicates that the angle is measured clockwise from the positive x-axis, the direction of the plane's velocity relative to the Earth is south of west.

Therefore, the direction of the plane's velocity relative to the Earth is "south of west."