An airplane is flying 400 km/h west. There is a wind blowing 100 km/h NE. What is the airplane's actual speed and heading?
To find the airplane's actual speed and heading, we need to combine its airspeed and the wind speed.
Let's first break down the given information:
- The airplane is flying 400 km/h west. This means the airplane's airspeed is 400 km/h in the western direction.
- There is a wind blowing 100 km/h NE (northeast). The wind is coming from the northeast, which means it is blowing toward the southwest.
Now, to find the actual speed and heading, we can use vector addition. We add the airspeed vector and the wind vector to determine the resultant vector, which represents the airplane's actual speed and direction.
Step 1: Draw a diagram:
- Start by drawing a horizontal line to represent the westward direction, with an arrowhead pointing to the left to show the airplane's airspeed.
- Then, draw a diagonal arrow from the northeast to the southwest to represent the wind blowing 100 km/h.
- Check that the angle between the airspeed vector and the wind vector is 45 degrees.
Step 2: Resolve the wind vector:
- Resolve the wind vector by splitting it into two perpendicular components.
- The vertical component represents the wind blowing downward from northeast to southwest, which we will call the south component (-100 km/h).
- The horizontal component represents the wind blowing from northeast to southwest, which we will call the west component (-100 km/h).
Step 3: Add the vectors:
- Add the horizontal components of the airspeed (400 km/h) and the wind (west component: -100 km/h).
- The resultant horizontal component is 400 km/h - 100 km/h = 300 km/h from west to east.
- Add the vertical components of the airspeed (zero, since the airplane is not moving vertically) and the wind (south component: -100 km/h).
- The resultant vertical component is 0 km/h - 100 km/h = -100 km/h from north to south.
Step 4: Calculate the magnitude and direction:
- Use the Pythagorean theorem to calculate the magnitude (actual speed) of the resultant vector.
- magnitude = sqrt((horizontal component)^2 + (vertical component)^2).
- magnitude = sqrt((300 km/h)^2 + (-100 km/h)^2).
- magnitude = sqrt(90000 km²/h² + 10000 km²/h²).
- magnitude = sqrt(100000 km²/h²).
- magnitude = 316.23 km/h.
Step 5: Calculate the direction:
- Use trigonometry to find the direction of the resultant vector.
- direction = arctan(vertical component / horizontal component).
- direction = arctan(-100 km/h / 300 km/h).
- direction = arctan(-1/3).
- direction ≈ -18.43 degrees.
Therefore, the airplane's actual speed is approximately 316.23 km/h, and its heading is approximately 18.43 degrees west of north.
Vpw = Vp + Vw = -400 + 100[45o] =
-400 + 100*Cos45 + 100*sin45 =
-400 + 70.7 + 70.7i=-329.3 + 70.7i(Q2) =
-337(-12.12o) = 337[-12.12+180] =
337km/h[168o]. = 337km/h[78o] W. of N.
break the NE into N, and E components. Then add to the W (W=-E)
Recombined into one vector.