An airplane is flying with an airspeed of 660 Kilometers per hour. The wind is blowing from 345 degrees at 35 km/h. What direction should the airplane take so that, with the wind, the plane will be flying in a direction of 72 degrees?
I can compute airspeed and actual direction using law of sines and cosines but not sure how to figure out the requested 72 degree part.
X = hor. = 660 + 35cos345 = km/h.
Y = ver. = 35sin345 =
tanA = Y/X = -9.06 / 693.8 = -.0131,
A = -0.75 deg.
A + B = 72,
-0.75 + B = 72,
B = 72 + 0.75 = 72.75 deg. = Direction.
Vpw = Vp + Vw = 660[72o].
Vp + 35[345o] = 660[72].
Vp+35*Cos345+35*sin345=660*Cos72+660*sin72.
Vp + 33.8-9.1i = 204+627.7i.
Vp=204 + 627.7i -33.8+9.1i=170.2+636.8i
= 659km/h[75o].
To find the direction the airplane should take, we can use vector addition.
1. The wind is blowing from 345 degrees at 35 km/h. To convert this to a vector form, we can break it into its horizontal and vertical components:
- Horizontal component: 35 km/h * cos(345 degrees)
- Vertical component: 35 km/h * sin(345 degrees)
Therefore, the wind vector is (-29.58, -22.31) km/h.
2. The plane wants to fly in a direction of 72 degrees relative to the ground when taking into account the wind. To convert this to a vector form, we can break it into its horizontal and vertical components:
- Horizontal component: 660 km/h * cos(72 degrees)
- Vertical component: 660 km/h * sin(72 degrees)
Therefore, the desired direction vector of the plane is (178.39, 635.72) km/h.
3. To find the direction the airplane should take, we need to subtract the wind vector from the desired direction vector:
- Horizontal component: 178.39 km/h - (-29.58 km/h)
- Vertical component: 635.72 km/h - (-22.31 km/h)
Therefore, the resulting vector is (207.97, 638.03) km/h.
4. Finally, we can find the direction of this resulting vector using the inverse tangent function:
Direction = atan(vertical component / horizontal component)
Direction = atan(638.03 km/h / 207.97 km/h)
Therefore, the direction the airplane should take is approximately 73.87 degrees.
So, to fly in a direction of 72 degrees when considering the wind, the airplane should take a direction of approximately 73.87 degrees.
To determine the direction the airplane should take, we can break down the problem into two components: the wind's effect on the airplane's speed and direction, and the desired direction of the plane.
First, let's find the actual speed and direction of the airplane. We can use vector addition to combine the airspeed vector and the wind vector.
To find the actual speed, we use the Pythagorean theorem:
Actual speed = √(airspeed^2 + wind speed^2)
= √(660^2 + 35^2)
≈ 663.38 km/h
To find the angle of the actual direction, we can use the inverse tangent (arctan) function. The angle can be calculated as:
Angle = arctan(wind speed / airspeed)
= arctan(35 / 660)
≈ 3.02 degrees
Now that we know the actual speed and direction of the airplane, we can proceed to find the direction it should take to achieve the desired direction of 72 degrees.
To find the angle between the airplane's actual direction and the desired direction, we subtract the actual direction angle from the desired direction angle:
Angle difference = desired direction - actual direction
= 72 - 3.02
≈ 68.98 degrees
So, the airplane should take a direction that is approximately 68.98 degrees clockwise (or counterclockwise depending on the reference frame) from its actual direction.