A plane is flying at a speed and direction of 550 mph, 25 degrees north of west. There is a wind of 28 mph, at an angle of 19 degrees south of the east direction. How fast is the plane actually flying? What direction is the plane traveling?

Question

A plane is flying at a speed and direction of 550 mph, 25 degrees north of west. There is a wind of 28 mph, at an angle of 19 degrees south of the east direction. How fast is the plane actually flying? What direction is the plane traveling?

Answer 1

Here is an easy way to do these.

1) convert all angles to 000N,090E, 180S, 270W

Example: 456@29deg S of E.
= 456@119
then convert to N, E coordinates.

V= 456(Cos119 N + Sin119 E)

Now for your problem:

V= 550(cos295 N +sin295 E)+28(cos109 N+ sin109)
combine N, E vector components:
V= N(550cos295+28cos109)+E(550sin295+28sin109)

do that math and you will have something like this
V= nnnn N + eeeee E

magnitude V= sqrt(nnnn^2+eeee^2)
direction= arctan (eeee/nnnn)
on the angle, be careful, as tangent is a repeating angle, it repeats each 180. So you may have to figure which quadrant.

Answer 2

To find the actual speed and direction of the plane, we can use vector addition.

First, let's break down the velocities of the plane and the wind into their components.

The velocity of the plane has two components: one towards the west and the other towards the north. We can find these components using trigonometry.

The velocity towards the west (Vx) can be calculated using the cosine function:
Vx = V * cos(θ)
Vx = 550 mph * cos(25°)

Similarly, the velocity towards the north (Vy) can be calculated using the sine function:
Vy = V * sin(θ)
Vy = 550 mph * sin(25°)

Next, let's determine the components of the wind velocity.

The velocity towards the east (Wx) can be calculated using the cosine function:
Wx = W * cos(θ)
Wx = 28 mph * cos(19°)

The velocity towards the south (Wy) can be calculated using the sine function:
Wy = W * sin(θ)
Wy = 28 mph * sin(19°)

Now, we can add the corresponding components together to determine the overall velocity of the plane.

The resulting velocity towards the west:
Vx - Wx

The resulting velocity towards the north:
Vy + Wy

To find the magnitude (the speed) of the plane, we can use the Pythagorean theorem:
Speed = sqrt((Vx - Wx)^2 + (Vy + Wy)^2)

Lastly, to find the direction of the plane, we can use the inverse tangent function:
Direction = atan((Vy + Wy) / (Vx - Wx))

By plugging in the given values into the formulas, you can calculate the final answer.