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. What was your solution to answering this problem?
Oh, flying planes, wind, and bearings... sounds like a real "high-flying" problem!
To find the resultant ground velocity of the plane, we can break it down into two components: the horizontal component (west-east direction) and the vertical component (south-north direction).
The horizontal component is the airspeed of 400 km/h, and it's heading on a bearing of 200°. So, using some fancy trigonometry, we can find that the horizontal component is 400 km/h * cos(200°).
The vertical component is given by the wind blowing at 100 km/h from the northeast. As the northeast direction is halfway between north and east (45°), we can calculate the vertical component as 100 km/h * sin(45°).
Now, to find the resultant ground velocity, we simply add the horizontal and vertical components together. And voila! You've got your answer.
To determine the resultant ground velocity of the plane, we can use vector addition. We'll break down the velocities into their components and then find the sum of the horizontal and vertical components separately.
1. The initial velocity of the plane can be broken down into its components:
- Horizontal component: 400 km/h * cos(200°)
- Vertical component: 400 km/h * sin(200°)
2. The velocity of the wind is blowing from the northeast, which is 45° from the east. So the wind velocity can be broken down into its components:
- Horizontal component: 100 km/h * cos(45°)
- Vertical component: 100 km/h * sin(45°)
3. The resultant velocity is obtained by adding the corresponding components:
- Horizontal component: (Horizontal component of plane's velocity) + (Horizontal component of wind velocity)
- Vertical component: (Vertical component of plane's velocity) + (Vertical component of wind velocity)
4. Finally, we can calculate the magnitude and direction of the resultant velocity using the Pythagorean theorem and trigonometry:
- Magnitude: square root of (Resultant horizontal component ^ 2 + Resultant vertical component ^ 2)
- Direction: arctan(Resultant vertical component / Resultant horizontal component)
By following these steps, we can determine the resultant ground velocity of the plane.
To find the resultant ground velocity of the plane, we need to consider the effects of both the plane's airspeed and the wind.
Here are the steps to solve the problem:
1. Draw a diagram: Draw a diagram representing the situation. Place the plane's initial heading (200°) and indicate the direction of the wind (northeast).
2. Resolve the wind velocity: Since the wind is coming from the northeast, we need to resolve its components into eastward and northward directions. The wind velocity can be split into two components: a northward component and an eastward component. The northward component can be calculated by multiplying the wind speed (100 km/h) by the sine of the angle between the wind direction and the north direction (45°). The eastward component can be calculated by multiplying the wind speed (100 km/h) by the cosine of the angle between the wind direction and the north direction (45°).
3. Calculate the resultant velocity: The resultant velocity is the vector sum of the plane's airspeed and the wind's velocity components. To calculate it, we need to add the eastward component of the wind velocity to the eastward component of the plane's airspeed and add the northward component of the wind velocity to the northward component of the plane's airspeed.
4. Find the direction and magnitude of the resultant velocity: The direction of the resultant velocity can be determined using trigonometry. The angle can be found by taking the inverse tangent of the northward component of the resultant velocity divided by the eastward component. The magnitude of the resultant velocity can be found using the Pythagorean theorem, by taking the square root of the sum of the squares of the eastward and northward components.
Based on the above steps, my solution to answering the problem would involve performing the calculations outlined and providing the direction and magnitude of the resultant ground velocity of the plane.