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.
I get
492.4 on a heading of 205°
To determine the resultant ground velocity of the plane, we need to consider the vector addition of the plane's air speed and the wind's speed. We will break down the velocities into their components and then add them together to find the resultant.
1. First, let's consider the airspeed of the plane. The bearing of 200° tells us the direction in which the plane is flying. We can interpret this as an angle with respect to the north, where 0° is north, 90° is east, 180° is south, and 270° is west. Since the bearing is 200°, it means the plane is flying southwest (200° is located between the south and west directions).
2. To determine the horizontal and vertical components of the airspeed, we can use trigonometry. Assuming that the positive x-axis points east and the positive y-axis points north, we can determine the x and y components as follows:
- Horizontal component (x): airspeed * cos(bearing)
x = 400 km/h * cos(200°)
- Vertical component (y): airspeed * sin(bearing)
y = 400 km/h * sin(200°)
3. Now, let's consider the wind velocity. The wind is blowing from the northeast. To find its components, we can represent the northeast direction as a bearing of 45° (45° is located between the north and east directions).
4. Use the same approach as before to determine the horizontal and vertical components of the wind velocity:
- Horizontal component (x): wind speed * cos(bearing)
x = 100 km/h * cos(45°)
- Vertical component (y): wind speed * sin(bearing)
y = 100 km/h * sin(45°)
5. Now that we have the horizontal and vertical components of both the airspeed and the wind velocity, we can add them together:
- Horizontal component (x) of resultant velocity: (airspeed x) + (wind x)
- Vertical component (y) of resultant velocity: (airspeed y) + (wind y)
6. Finally, we can calculate the magnitude (speed) and direction of the resultant velocity using the Pythagorean theorem and trigonometry:
- Resultant speed: √[(resultant x)^2 + (resultant y)^2]
- Resultant direction with respect to north: arctan(resultant y / resultant x)
By following these steps, you should be able to determine the resultant ground velocity of the plane given its airspeed and the wind velocity.