A plane is flying due south at a speed of 300 mph. The wind is blowing from the west at a speed of 40 mph. Find the resultant speed of the plane and the direction of the plane

V = 300 mi/h @ 270Deg + 40 mi/h @ 0Deg.

X = 40 mi/h.
Y = -300 mi/h.

tanA = Y/X = -300 / 40 = -7.500.
A = -82.4 Deg.,CW.
A = -82.4 + 360 = 277.6 Deg, CCW.

Vp = X/cosA = 40 / cos277.6 = 303 mi/h
@ 277.6 Deg.

X=68+1

To find the resultant speed of the plane, we need to use vector addition. We'll use the Pythagorean theorem to find the magnitude of the resultant velocity and trigonometry to find the direction.

Let's break down the velocities into their respective components. Since the plane is moving due south, the magnitude of its velocity is 300 mph, and its y-component is -300 mph. Similarly, the wind is blowing from the west, so the magnitude of its velocity is 40 mph, and its x-component is -40 mph.

To find the resultant velocity, we add the x- and y-components separately.

x-component (resultant_x) = plane_x + wind_x = 0 + (-40) = -40 mph
y-component (resultant_y) = plane_y + wind_y = -300 + 0 = -300 mph

Now, let's find the magnitude of the resultant velocity using the Pythagorean theorem:

resultant_speed = √(resultant_x^2 + resultant_y^2)
= √((-40)^2 + (-300)^2)
= √(1600 + 90000)
= √91600
≈ 302.66 mph

The resultant velocity speed is approximately 302.66 mph.

Lastly, we can find the direction of the resultant velocity using trigonometry. The direction can be determined by finding the angle θ, where:

θ = arctan(resultant_y / resultant_x)
= arctan(-300 / -40)
= arctan(7.5)
≈ 80.54°

The direction of the plane is approximately 80.54°.

Therefore, the resultant speed of the plane is approximately 302.66 mph, and the direction of the plane is approximately 80.54°.

To find the resultant speed of the plane, we will use vector addition. We can break down the velocity of the plane and the velocity of the wind into their horizontal and vertical components.

Given:
- The plane is flying due south, which means its velocity component in the vertical direction is 300 mph.
- The wind is blowing from the west, which means its velocity component in the horizontal direction is 40 mph.

To find the resultant velocity in the vertical direction, we don't need to consider the wind since it only affects the horizontal direction. Therefore, the vertical component of the plane's velocity remains at 300 mph.

To find the resultant velocity in the horizontal direction, we can subtract the velocity of the wind (from the west) from the velocity of the plane (flying due south). Since they are perpendicular to each other, we can use the Pythagorean theorem to find the hypotenuse, which represents the magnitude of the resultant velocity.

Using Pythagorean theorem:
(Resultant velocity)^2 = (velocity of the plane)^2 + (velocity of the wind)^2
(Resultant velocity)^2 = 300^2 + 40^2
(Resultant velocity)^2 = 90000 + 1600
(Resultant velocity)^2 = 91600

Taking the square root of both sides, we get:
Resultant velocity = sqrt(91600)
Resultant velocity ≈ 302.7 mph

Now, to determine the direction of the plane, we need to find the angle it makes with the due south direction. We can use trigonometry to do this.

The tangent of the angle is given by the opposite side (velocity of the wind) divided by the adjacent side (velocity of the plane).

Tangent of the angle = (velocity of the wind) / (velocity of the plane)
Tangent of the angle = 40 / 300 ≈ 0.1333

Taking the inverse tangent (or arctan) of 0.1333, we can find the angle.

Angle ≈ arctan(0.1333)
Angle ≈ 7.613 degrees

So, the resultant speed of the plane is approximately 302.7 mph, and the direction of the plane is approximately 7.613 degrees west of due south.