An airplane travels on a bearing of 245 degrees at an airspeed of 200 kilometers per hour while a wind is blowing 50 kilometers per hour from 140 degrees. Find the ground speed of the airplane and the direction of its track, or course, over the ground.
First, vessels travel on headings, not bearings.
200 @ 245° = (-181.262,-84.524)
50 @ 320° = (-32.134,38.302)
add them up to get
(-213.396,-46.222) = 218 @ 258°
To find the ground speed and direction of the airplane's track, we need to break down the problem into its components: the airplane's velocity component and the wind's velocity component. Then we can add them together to find the resultant velocity.
1. First, let's find the x and y components of the airplane's velocity.
- The given bearing of 245 degrees indicates the direction of the airplane's heading relative to true north.
- To find the x and y components, we can use trigonometry.
- The x component, Vx, is given by V * cos(θ), where V is the airspeed and θ is the bearing.
- The y component, Vy, is given by V * sin(θ), where V is the airspeed and θ is the bearing.
We have:
- V = 200 km/h (airspeed)
- θ = 245 degrees (bearing)
So, Vx = V * cos(θ) = 200 km/h * cos(245°)
And Vy = V * sin(θ) = 200 km/h * sin(245°)
2. Next, let's find the x and y components of the wind's velocity.
- The given wind speed, 50 km/h, is not a bearing relative to true north. So we need to convert it into x and y components.
- To find the components, we can use trigonometry.
- The x component, Wx, is given by W * cos(θ), where W is the wind speed and θ is the angle at which the wind is blowing (measured relative to true north).
- The y component, Wy, is given by W * sin(θ), where W is the wind speed and θ is the angle at which the wind is blowing (measured relative to true north).
We are given:
- W = 50 km/h (wind speed)
- θ = 140 degrees (angle at which the wind is blowing)
So, Wx = W * cos(θ) = 50 km/h * cos(140°)
And Wy = W * sin(θ) = 50 km/h * sin(140°)
3. Now, let's find the resultant velocity (Vr) by adding the x and y components of both the airplane's velocity and the wind's velocity.
- Vr = (Vx + Wx, Vy + Wy)
Plugging in the values we calculated earlier, we can find the resultant velocity.
4. Finally, let's find the ground speed and direction of the airplane:
- The ground speed is simply the magnitude of the resultant velocity.
- Ground Speed = sqrt((Vr_x)^2 + (Vr_y)^2)
- The direction of the track/course over the ground can be found using trigonometry.
- Track Direction = arctan(Vr_y / Vr_x), where arctan is the inverse tangent function.
By following these steps, we can find the ground speed of the airplane and the direction of its track over the ground.