An airplane is flying at a velocity of 300km/h S30degW in a wind blowing at a velocity of 50km/h S20degE. What is the true speed and direction of the plane?

334 km/h @ S23.4W

To find the true speed and direction of the plane, we need to add the velocities of the plane and the wind vectorially. We'll begin by breaking down the velocities into their horizontal and vertical components.

Given:
Velocity of plane = 300 km/h at S30degW
Velocity of wind = 50 km/h at S20degE

Step 1: Resolve the velocities into horizontal and vertical components.

For the velocity of the plane:
- The horizontal component = velocity * cos(angle)
- The vertical component = velocity * sin(angle)

Velocity of plane:
Horizontal component = 300 km/h * cos(30deg)
= 300 km/h * cos(30deg)
= 300 km/h * (√3/2)
= 259.8076 km/h

Vertical component = 300 km/h * sin(30deg)
= 300 km/h * 1/2
= 150 km/h

For the velocity of the wind:
- The horizontal component = velocity * cos(angle)
- The vertical component = velocity * sin(angle)

Velocity of wind:
Horizontal component = 50 km/h * cos(20deg)
= 50 km/h * cos(20deg)
= 50 km/h * (9√3/10)
= 43.3013 km/h

Vertical component = 50 km/h * sin(20deg)
= 50 km/h * (1/2)
= 25 km/h

Step 2: Add the horizontal components of the plane and the wind.
True horizontal speed = Horizontal component of plane + Horizontal component of wind
= 259.8076 km/h + 43.3013 km/h
= 303.1089 km/h

Step 3: Add the vertical components of the plane and the wind.
True vertical speed = Vertical component of plane + Vertical component of wind
= 150 km/h + 25 km/h
= 175 km/h

Step 4: Use the Pythagorean theorem to find the true speed of the plane.
True speed = √(horizontal speed^2 + vertical speed^2)
= √(303.1089 km/h^2 + 175 km/h^2)
= √(92100.0481 + 30625)
= √122725.0481
= 350.2 km/h (rounded to the nearest tenth)

Step 5: Use trigonometry to find the direction of the plane.
True direction = atan(vertical speed / horizontal speed)
= atan(175 km/h / 303.1089 km/h)
= atan(0.5770)
= 29.8181deg (rounded to the nearest tenth)

Therefore, the true speed of the plane is approximately 350.2 km/h, and its true direction is approximately S29.8degW.

To find the true speed and direction of the plane, we need to use vector addition and trigonometry.

1. First, let's decompose the airplane's velocity into its components. The given velocity of 300 km/h S30°W can be expressed as follows:
- The north-south component (y-component) is the opposite side of the 30° angle, given by: Vplane_y = 300 km/h * sin(30°).
- The east-west component (x-component) is the adjacent side of the 30° angle, given by: Vplane_x = 300 km/h * cos(30°).

2. Similarly, decompose the wind velocity of 50 km/h S20°E into its components:
- The north-south component (y-component) is the opposite side of the 20° angle, given by: Vwind_y = 50 km/h * sin(20°).
- The east-west component (x-component) is the adjacent side of the 20° angle, given by: Vwind_x = 50 km/h * cos(20°).

3. Add the x-components and y-components separately to find the resultant components. The resultant north-south component is given by: R_y = Vplane_y + Vwind_y, and the resultant east-west component is given by: R_x = Vplane_x + Vwind_x.

4. Calculate the magnitude of the resultant vector (true speed of the plane) using the Pythagorean theorem:
R = sqrt(R_x^2 + R_y^2).

5. Find the direction of the resultant vector (true direction of the plane) using trigonometry:
- Tanθ = R_y / R_x.
- θ = atan2(R_y, R_x).

6. Convert the direction angle to compass notation. Since the given value is S30°W, we need to convert it to a positive angle in standard compass notation:
- True direction = (180° - θ) + 180°.

Using these steps, we can solve for the true speed and direction of the plane.

(Note: In the calculations, angles are measured in radians.