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A plane is heading on a bearing of 200° with an air speed of 400 km/h when it is blown off course by a wind of 100 km/h from the northeast. Determine the resultant ground velocity of the plane.

Math teachers do not know navigation English. It drives me crazy.

You do not head on a bearing.
You sail or fly on a heading.

the plane is flying on a heading of 200 (about SSW) at 400 km/h

the plane is also moving at 100 km/h southwest which is 180+45 = 225 degrees

You can do this with components south and west or you can solve a parallelogram

The component way:
speed south = 200 cos 20 + 100 cos 45
= 259 south

speed west = 200 sin 20 + 100 sin 45
= 139 west
so
speed = sqrt (259^2 +139^2)
= 294 km/hr
tan angle west of south = 139/259
angle west of south = 28 degrees
180 + 28 = 208 degrees
so
velocity wrt ground = 294 km/h at 208 degrees clockwise from North

"bearing" means what direction you LOOK.

"heading" means what direction you point the vehicle.

As in: We were heading north when we took a bearing of 90 degrees on a lighthouse east of us.

To determine the resultant ground velocity of the plane, we can break down the problem into components.

Step 1: Find the components of the wind velocity.
Since the wind is blowing from the northeast, we can determine its components using trigonometry. Recall that the northeast direction forms a 45° angle with both the east and north directions.

The east component of the wind velocity can be found by multiplying the wind speed (100 km/h) by the cosine of the angle between the east direction and the northeast direction (45°).
East component = 100 km/h * cos(45°)

The north component of the wind velocity can be found by multiplying the wind speed (100 km/h) by the sine of the angle between the north direction and the northeast direction (45°).
North component = 100 km/h * sin(45°)

Step 2: Find the components of the resultant velocity.
Given that the plane is heading on a bearing of 200° with an airspeed of 400 km/h, we can determine its components. We'll use the same trigonometry principles from Step 1.

The east component of the resultant velocity can be found by multiplying the airspeed (400 km/h) by the cosine of the angle between the east direction and the bearing of 200°.
East component = 400 km/h * cos(200°)

The north component of the resultant velocity can be found by multiplying the airspeed (400 km/h) by the sine of the angle between the north direction and the bearing of 200°.
North component = 400 km/h * sin(200°)

Step 3: Find the sum of the east and north components.
To find the resultant ground velocity, we need to add the east components and the north components together.

Resultant east component = East component of wind velocity + East component of resultant velocity
Resultant north component = North component of wind velocity + North component of resultant velocity

Step 4: Calculate the magnitude and direction of the resultant ground velocity.
To calculate the magnitude of the resultant ground velocity, we can use the Pythagorean theorem. The magnitude represents the speed, while the direction represents the bearing of the velocity.

Magnitude = √(Resultant east component^2 + Resultant north component^2)
Direction = arctan(Resultant east component / Resultant north component)

By following these steps, you can determine the resultant ground velocity of the plane.

To determine the resultant ground velocity of the plane, you need to break down the given information and apply vector addition.

1. Start by drawing a diagram to represent the situation. Plot the initial heading of the plane (bearing of 200°) and label the airspeed (400 km/h).

2. Next, draw a vector representing the wind blowing from the northeast. Since it is given as a wind speed of 100 km/h, you can represent it as a magnitude and direction.

3. To determine the direction of the wind, note that "northeast" is at a 45° angle from both the north and east. Therefore, the wind direction is 45° south of east.

4. Convert the bearing of the plane to a vector direction by subtracting it from 90° (since north is 90°). So, 90° - 200° = -110°. This means the plane is heading 110° south of east.

5. Now, you need to determine the components (magnitude and direction) of the airspeed and the wind velocity in terms of east-west and north-south directions.

- For the airspeed: Since the plane is heading 110° south of east, you can break down the airspeed into its vector components by using trigonometry. The east-west component (Ve) is given by cos(110°) × 400 km/h, and the north-south component (Vn) is given by sin(110°) × 400 km/h.

- For the wind velocity: Since the wind direction is 45° south of east, you can break down the wind velocity into its vector components. The east-west component (We) is given by cos(45°) × 100 km/h, and the north-south component (Wn) is given by sin(45°) × 100 km/h.

6. Now, add the vector components for the airspeed and the wind velocity to obtain the resultant ground velocity vector.

- The east-west component of the resultant ground velocity: Ve + We
- The north-south component of the resultant ground velocity: Vn + Wn

7. Finally, calculate the magnitude and direction of the resultant ground velocity vector using these components. The magnitude can be determined using the Pythagorean theorem (magnitude = √(east-west component^2 + north-south component^2)), and the direction can be determined using trigonometry.

By following these steps, you can determine the resultant ground velocity of the plane.