An airplane is flying at 550 km/h on a heading of 080 degrees. The wind is blowing at 60 km/h from a bearing of 120 degrees. Find the ground velocity of the airplane and include the new bearing.
This is not calculus.
The air (wind) velocity PLUS the plane's air velocity equals the ground velocity.
Perform the vector addition
Note: Vpw = 597km/h[83.7o] E. of N.
To find the ground velocity of the airplane, we need to find the resultant vector by considering the airplane's velocity and the wind's velocity.
First, let's break down the airplane's velocity into its components. The airplane's heading is given as 080 degrees, which means it is pointing 80 degrees east of north. We can represent this as a vector with an eastward component (x-axis) and a northward component (y-axis).
The eastward component can be found using trigonometry. We can find the x-component by using the cosine function, since the cosine of an angle gives us the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side represents the eastward component, and the hypotenuse represents the magnitude of the airplane's velocity.
The northward component can be found using the sine function, since the sine of an angle gives us the ratio of the opposite side to the hypotenuse. In this case, the opposite side represents the northward component, and the hypotenuse represents the magnitude of the airplane's velocity.
Let's calculate the airplane's eastward and northward components:
Eastward component (Vx) = Velocity * cos(heading)
= 550 km/h * cos(80 degrees)
Northward component (Vy) = Velocity * sin(heading)
= 550 km/h * sin(80 degrees)
Now let's break down the wind's velocity into its components. The wind is blowing at 60 km/h from a bearing of 120 degrees, which means it is coming from the southwest. We can represent this as a vector with a westward component (x-axis) and a southward component (y-axis).
Since the wind is blowing from the southwest, we need to find the westward and southward components. We can use the same approach as before, using the cosine function to find the westward component and the sine function to find the southward component.
Westward component of wind = Wind speed * cos(180 - bearing)
= 60 km/h * cos(180 - 120 degrees)
Southward component of wind = Wind speed * sin(180 - bearing)
= 60 km/h * sin(180 - 120 degrees)
Now we can calculate the resultant vector by summing up the corresponding components:
Resultant eastward component = airplane's eastward component + wind's westward component
= Vx - (wind's westward component)
Resultant northward component = airplane's northward component + wind's southward component
= Vy - (wind's southward component)
Now that we have the resultant vector's eastward component and northward component, we can find its magnitude and angle using the Pythagorean theorem and inverse tangent respectively.
Magnitude of the resultant vector (ground velocity) = sqrt(Resultant eastward component^2 + Resultant northward component^2)
Angle (bearing) of the resultant vector = atan2(Resultant eastward component, Resultant northward component)
By calculating these values, you will find the ground velocity of the airplane and the new bearing.
Vpw = Vp + Vw = 550[80o] + 60[120o] =
550*cos80+550*sin80 +60*Cos120+60*sin120
= 95.5+541.6i - 30+52i = 65.5 + 593.6i.
= 597mi/h[83.7o].