a pilot wishes to fly on a course bearing 45 and with a ground speed of 600 km per hour. If a wind is blowing from the southwest (bearing 315) at 80 km per hour. What must be the heading and air speed of the aircraft?
Vp + 80km/h[315] = 600[45o].
Vp + 80*sin315 + i80*Cos315 = 600*sin45 + i600*Cos45,
Vp - 56.6 + i56.6 = 424.3 + i424.3,
Vp = 480.9 + i367.7 = 605.4km/h[52.6o] = Air speed and heading of the plane.
Vp + 80[45o] = 600[45o].
Vp + 80*sin45 +i80*Cos45 = 600*sin45 + i600*Cos45,
Vp + 56.6 + i56.6 = 424.3 + i424.3,
Vp = 367.7 + i367.7 = 520 km/h[45o]. = Air speed and heading of the plane.
Note: If the wind is blowing from southwest, the bearing is 225o Not 315o.
To find the heading and airspeed of the aircraft, we need to account for the effect of wind on the plane's actual speed and direction. This can be done through vector addition.
First, let's break down the wind speed into its components along the north-south and east-west axes.
The wind bearing is 315 degrees, which means it is blowing from the southwest. To find the components, we use trigonometry.
The component blowing northward: Wind speed * sin(180 - wind bearing)
Component north = 80 * sin(180 - 315)
The component blowing eastward: Wind speed * cos(180 - wind bearing)
Component east = 80 * cos(180 - 315)
Now, let's calculate the components of the plane's velocity along the north-south and east-west axes using the ground speed and course bearing values.
The component flying northward: Ground speed * sin(180 - course bearing)
Component north = 600 * sin(180 - 45)
The component flying eastward: Ground speed * cos(180 - course bearing)
Component east = 600 * cos(180 - 45)
To find the total northward and eastward components, we add the plane's and wind's components:
Total component north = Component north (plane) + Component north (wind)
Total component east = Component east (plane) + Component east (wind)
Finally, we can find the resultant of the plane's velocity by using the Pythagorean theorem:
Resultant velocity = √(Total component north)^2 + (Total component east)^2
The magnitude of the resultant vector represents the airspeed of the aircraft, and the angle it makes with the north direction is the heading.
To summarize the steps to find the heading and airspeed of the aircraft:
1. Calculate the components of the wind vector.
2. Calculate the components of the plane's velocity.
3. Calculate the total northward and eastward components.
4. Use the Pythagorean theorem to find the resultant velocity.
5. The magnitude of the resultant is the airspeed of the aircraft, and the angle it makes with the north direction is the heading.