An airplane is flying at 550 km/h on a heading of 080 degrees: The wind is blowing at 60 km/h from a bearing of 120 degrees.

Find the ground velocity of the airplane.

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To find the ground velocity of the airplane, we need to use vector addition. Let's break down the given information into components:

Airplane speed (air speed): 550 km/h
Airplane heading: 080 degrees
Wind speed: 60 km/h
Wind bearing: 120 degrees

First, let's convert the given angles to radians because trigonometric functions in mathematics usually work with radians.

Airplane heading in radians = 080 degrees × π/180 = 1.396 radians
Wind bearing in radians = 120 degrees × π/180 = 2.094 radians

Next, we can determine the components of the airplane's speed and wind speed:

Horizontal component of airplane speed = Airplane speed × cos(Airplane heading)
Vertical component of airplane speed = Airplane speed × sin(Airplane heading)
Horizontal component of wind speed = Wind speed × cos(Wind bearing)
Vertical component of wind speed = Wind speed × sin(Wind bearing)

Now, let's calculate the components:

Horizontal component of airplane speed = 550 km/h × cos(1.396 radians) ≈ 512.574 km/h
Vertical component of airplane speed = 550 km/h × sin(1.396 radians) ≈ 198.895 km/h
Horizontal component of wind speed = 60 km/h × cos(2.094 radians) ≈ -30 km/h (negative because it's blowing in the opposite direction)
Vertical component of wind speed = 60 km/h × sin(2.094 radians) ≈ 51.961 km/h

Finally, we can find the total ground velocity of the airplane by adding the horizontal components and the vertical components separately:

Horizontal component of ground velocity = Horizontal component of airplane speed + Horizontal component of wind speed
= 512.574 km/h + (-30 km/h) ≈ 482.574 km/h
Vertical component of ground velocity = Vertical component of airplane speed + Vertical component of wind speed
= 198.895 km/h + 51.961 km/h ≈ 250.856 km/h

The total ground velocity of the airplane can be found using the Pythagorean theorem:

Ground velocity = √(Horizontal component of ground velocity^2 + Vertical component of ground velocity^2)
= √(482.574 km/h^2 + 250.856 km/h^2)
≈ √343721.1 km^2/h^2
≈ 586.419 km/h

Therefore, the ground velocity of the airplane is approximately 586.419 km/h.

To find the ground velocity of the airplane, we need to consider the effect of both the airplane's velocity and the wind's velocity.

The airplane's velocity, also known as the airspeed, is given as 550 km/h at a heading of 080 degrees.

The wind's velocity is given as 60 km/h from a bearing of 120 degrees.

To determine the ground velocity, we need to consider the vector addition of the airplane's velocity and the wind's velocity. This can be done using vector addition techniques, such as the head-to-tail method or the component method.

Using the component method, we can break down the velocities into their respective north-south and east-west components.

For the airplane's velocity:
- The north-south component can be calculated using the formula: airspeed * sin(heading)
- The east-west component can be calculated using the formula: airspeed * cos(heading)

For the wind's velocity:
- The north-south component can be calculated using the formula: wind speed * sin(wind bearing)
- The east-west component can be calculated using the formula: wind speed * cos(wind bearing)

Adding up the north-south and east-west components of both the airplane and wind velocities will give us the ground velocity of the airplane.

Let's now calculate the ground velocity step by step:

First, calculate the north-south component of the airplane's velocity:
airspeed * sin(heading) = 550 km/h * sin(80 degrees)

Next, calculate the east-west component of the airplane's velocity:
airspeed * cos(heading) = 550 km/h * cos(80 degrees)

Then, calculate the north-south component of the wind's velocity:
wind speed * sin(wind bearing) = 60 km/h * sin(120 degrees)

Finally, calculate the east-west component of the wind's velocity:
wind speed * cos(wind bearing) = 60 km/h * cos(120 degrees)

Now, add up the north-south components of the airplane's and wind's velocities, and similarly, add up the east-west components of both velocities. The resulting sums will give us the ground velocity of the airplane.