An airplane flies due north at 240 km/h relative to the air. There is a wind blowing at 60 km/h to the northeast relative to the ground. What are the plane's speed and direction relative to the ground?
Answers to Find:
km/h = ?
° east of north = ?
All angles are measured CW from +Y-axis.
Vr = 240km/h[0o] + 60km[45o].
X = 240*sin 0 + 60*sin45 = 42.4km/h.
Y = 240*Cos 0 + 60*Cos45 = 282.4 km/h
Vr = sqrt (X^2 + Y^2) = Resultant velocity.
Tan A = X/Y, A = Degrees CW = Direction.
=
To find the plane's speed and direction relative to the ground, we can use vector addition. Let's break down the velocities into their north and east components.
Given:
Plane's velocity relative to the air (Vp-air) = 240 km/h due north
Wind's velocity relative to the ground (Vw-ground) = 60 km/h to the northeast
Step 1: Resolve the plane's velocity relative to the air (Vp-air) into north and east components.
The north component (Vp-air,north) will be equal to the magnitude of the plane's velocity relative to the air (240 km/h) since the plane is flying due north.
Vp-air,north = 240 km/h
The east component (Vp-air,east) will be 0 km/h since the plane is not flying in the east-west direction.
Vp-air,east = 0 km/h
Step 2: Resolve the wind's velocity relative to the ground (Vw-ground) into north and east components.
We can use basic trigonometry to find the north and east components of the wind's velocity.
The north component (Vw-ground,north) can be found by multiplying the magnitude of the wind's velocity relative to the ground (60 km/h) by the cosine of 45 degrees (since the angle between northeast and north is 45 degrees).
Vw-ground,north = 60 km/h * cos(45°)
Vw-ground,north = 60 km/h * 0.707
Vw-ground,north ≈ 42.42 km/h
The east component (Vw-ground,east) can be found by multiplying the magnitude of the wind's velocity relative to the ground (60 km/h) by the sine of 45 degrees (since the angle between northeast and east is 45 degrees).
Vw-ground,east = 60 km/h * sin(45°)
Vw-ground,east = 60 km/h * 0.707
Vw-ground,east ≈ 42.42 km/h
Step 3: Add the north and east components of the plane's velocity relative to the air (Vp-air) and the wind's velocity relative to the ground (Vw-ground) to find the resulting velocity relative to the ground (Vresultant).
The north component (Vresultant,north) will be the sum of the north components of Vp-air and Vw-ground.
Vresultant,north = Vp-air,north + Vw-ground,north
Vresultant,north = 240 km/h + 42.42 km/h
Vresultant,north ≈ 282.42 km/h
The east component (Vresultant,east) will be the sum of the east components of Vp-air and Vw-ground.
Vresultant,east = Vp-air,east + Vw-ground,east
Vresultant,east = 0 km/h + 42.42 km/h
Vresultant,east ≈ 42.42 km/h
Step 4: Use the north and east components of the resulting velocity relative to the ground (Vresultant,north and Vresultant,east) to find the speed and direction relative to the ground.
The speed (km/h) can be found using the magnitude of Vresultant.
Speed = √(Vresultant,north^2 + Vresultant,east^2)
Speed = √(282.42 km/h)^2 + (42.42 km/h)^2
Speed ≈ √(79785.57 km^2/h^2 + 1796.07 km^2/h^2)
Speed ≈ √(81581.64 km^2/h^2)
Speed ≈ 285.67 km/h
The direction (° east of north) can be found using the arctangent of the ratio of the east component (Vresultant,east) to the north component (Vresultant,north).
° east of north = arctan(Vresultant,east / Vresultant,north)
° east of north = arctan(42.42 km/h / 282.42 km/h)
° east of north ≈ 8.59°
Therefore, the plane's speed relative to the ground is approximately 285.67 km/h, and it is heading approximately 8.59° east of north.
To find the plane's speed and direction relative to the ground, we need to combine the velocity of the plane relative to the air with the velocity of the wind relative to the ground.
Let's break down the given information:
1. The plane flies due north at 240 km/h relative to the air.
2. There is a wind blowing at 60 km/h to the northeast relative to the ground.
First, let's find the resultant velocity of the plane relative to the ground, which is the combination of the plane's velocity relative to the air and the wind's velocity relative to the ground.
To find the resultant velocity, we can use vector addition. In this case, since the direction of the plane's velocity is due north and the wind's velocity is to the northeast, we will need to break down the wind's velocity into its north and east components.
Given that the wind velocity is 60 km/h, and it is blowing to the northeast, we can determine the north and east components using trigonometry.
The northeast direction can be divided into north and east directions at a 45-degree angle. Since the north component and the east component are equal, we can use the Pythagorean theorem to find each component.
Using the equation: (north component)^2 + (east component)^2 = (wind velocity)^2
Let's calculate the north and east components:
North Component:
Using the Pythagorean theorem, the north component is √[(60 km/h)^2 / 2]
North Component = √(3600 km²/h² / 2) ≈ √(1800 km²/h²) ≈ 42.4 km/h
East Component:
Using the Pythagorean theorem, the east component is √[(60 km/h)^2 / 2]
East Component = √(3600 km²/h² / 2) ≈ √(1800 km²/h²) ≈ 42.4 km/h
Now, we can combine the plane's velocity (due north at 240 km/h) with the wind's velocity (north component = 42.4 km/h and east component = 42.4 km/h).
To find the resultant velocity, we can add each component separately:
Resultant North Component = Plane's velocity + Wind's north component
= 240 km/h + 42.4 km/h
= 282.4 km/h
Resultant East Component = Wind's east component
= 42.4 km/h
Now we have the north and east components of the resultant velocity. To find the magnitude (speed) and the direction relative to the ground, we can use the Pythagorean theorem and trigonometry.
Magnitude (Speed):
Using the Pythagorean theorem, the magnitude (speed) is √[(north component)^2 + (east component)^2]
Magnitude = √[(282.4 km/h)^2 + (42.4 km/h)^2]
= √(79738 + 1798.56)
= √(81536.56)
≈ 285.63 km/h
Direction (° east of north):
To find the direction, we can use the inverse tangent (arctan) function with the east and north components.
Direction (° east of north) = arctan(east component / north component)
= arctan(42.4 km/h / 282.4 km/h)
≈ 7.62°
Therefore, the plane's speed relative to the ground is approximately 285.63 km/h, and it is approximately 7.62° east of north.
resolve the wind into N and E components
add them to the plane's N and E components
use the resulting components to find the plane's speed and direction
a diagram may prove helpful