A plane wants to fly east. It is capable of flying at 250 m/s, yet a 50 m/s wind is blowing [SW]. What heading must the plane set and what will be the plane's speed relative to the ground?
the plane must fly northeasterly so that its north velocity component equals the south velocity component of the wind
250 sin(Θ) = 50 sin(45º)
To determine the heading the plane must set, we need to consider the direction of the wind and the desired direction of flight.
Given that the wind is blowing from the southwest ([SW]), we can calculate the angle between the wind direction and the plane's desired flight direction.
Using trigonometry, we can find this angle:
tan(θ) = opposite/adjacent
tan(θ) = 50/250
θ = arctan(50/250) = 11.31 degrees
Therefore, the plane must set a heading of 11.31 degrees east of due north to counteract the wind blowing from the southwest.
To calculate the plane's speed relative to the ground, we can use vector addition.
The plane's desired speed is 250 m/s east, and the wind is blowing at 50 m/s southwest.
To find the resultant velocity (speed relative to the ground), we subtract the wind velocity vector from the plane's velocity vector:
Resultant velocity = sqrt((250^2) + (50^2)) = sqrt(62500 + 2500) = sqrt(65000) ≈ 254.95 m/s
Therefore, the plane's speed relative to the ground will be approximately 254.95 m/s.
To find out the heading and speed of the plane relative to the ground, we need to consider the effect of the wind.
Let's break down the motion into components:
1. The plane's speed along its intended direction of east is 250 m/s.
2. The wind speed in the southwest (SW) direction is 50 m/s.
Now, we can break down these speeds into their eastward (x) and northward (y) components using trigonometry.
Since the wind is blowing in the southwest direction, which is 45 degrees south of west, we need to find the eastward and northward components of the wind speed using basic trigonometry.
Given:
Plane speed (Vp) = 250 m/s
Wind speed (Vw) = 50 m/s
Angle of Wind (θ) = 45 degrees (south of west)
To find the eastward component, we use the cosine of the angle:
Vw_x = Vw * cos(θ)
Substituting the values:
Vw_x = 50 m/s * cos(45)
Using the value of cos(45) ≈ 0.707:
Vw_x = 50 m/s * 0.707 ≈ 35.355 m/s
To find the northward component, we use the sine of the angle:
Vw_y = Vw * sin(θ)
Substituting the values:
Vw_y = 50 m/s * sin(45)
Using the value of sin(45) ≈ 0.707:
Vw_y = 50 m/s * 0.707 ≈ 35.355 m/s
Now, we can calculate the resultant velocity (Vr) by adding the respective eastward and northward components:
Vr_x = Vp - Vw_x
Vr_y = -Vw_y (Note the negative sign indicating a southward direction)
Substituting the values:
Vr_x = 250 m/s - 35.355 m/s ≈ 214.645 m/s
Vr_y = -35.355 m/s
The resultant velocity (Vr) is given by:
Vr = sqrt(Vr_x^2 + Vr_y^2)
Substituting the values:
Vr = sqrt((214.645)^2 + (-35.355)^2) ≈ 218.492 m/s
To find the heading of the plane relative to the ground, we use the tangent of the angle:
tan(θ) = Vr_y / Vr_x
Substituting the values:
tan(θ) = -35.355 m/s / 214.645 m/s
Taking the inverse tangent (tan^(-1)) of both sides:
θ = tan^(-1)(-35.355 / 214.645)
Using a calculator or trigonometric table:
θ ≈ -9.37 degrees
Therefore, the plane should set a heading of approximately 9.37 degrees south of east. The plane's speed relative to the ground will be approximately 218.492 m/s.