An airplane is heading due south at a speed of 734 km/h . A wind begins blowing from the southwest at a speed of 110 km/h (average).
In what direction should the pilot aim the plane so that it will fly due south?
The heading of wind from southwest is
northeast which is 45o N. of E.
Vaw = Va + Vw = -734i.
Va + 110[45o] = -734i.
Va + 110*Cos45 + i110*sin45 = -734i.
Va + 77.8 + 77.8i = -734i.
Va = -77.8 - 811.8i, Q3.
Tan A = -811.8/-77.8. = 10.43445.
A = 84.5o S. of W. = 5.5o W. of S. =
The direction.
To find the direction the pilot should aim the plane so that it will fly due south, we need to consider the effect of the wind on the plane's flight path.
First, let's break down the velocities of the plane and the wind into their horizontal (east-west) and vertical (north-south) components. The plane is heading due south, so its velocity component in the north-south direction is 734 km/h.
Now, let's determine the components of the wind's velocity. The wind is blowing from the southwest, which is 45 degrees west of south. We can use trigonometry to find its north-south and east-west components.
The wind's velocity component in the north-south direction is given by sin(45 degrees) * 110 km/h ≈ 77.8 km/h (negative because it's blowing north).
The wind's velocity component in the east-west direction is given by cos(45 degrees) * 110 km/h ≈ 77.8 km/h (positive because it's blowing east).
To determine the net effect of the wind on the plane's flight path, we subtract the wind's velocity components from the plane's velocity components.
The net velocity component in the north-south direction is 734 km/h - 77.8 km/h = 656.2 km/h (southward).
The net velocity component in the east-west direction is 0 km/h + 77.8 km/h = 77.8 km/h (eastward).
Therefore, the pilot should aim the plane slightly eastward (towards the east) so that it will fly due south, compensating for the effect of the wind.
To determine the direction in which the pilot should aim the plane, we need to consider the vector addition of the airplane's velocity and the wind's velocity.
1. First, let's resolve the wind's velocity into its north and east components. Since the wind is blowing from the southwest, it forms a 45-degree angle with the south direction.
Using trigonometry, we can calculate the north and east components of the wind's velocity:
North component = wind speed * sin(45°)
= 110 km/h * sin(45°)
≈ 77.78 km/h
East component = wind speed * cos(45°)
= 110 km/h * cos(45°)
≈ 77.78 km/h
2. Now, let's consider the direction in which the plane is heading. The plane is heading due south, so its velocity vector only has a south component, which is equal to its speed.
Plane's south component = 734 km/h
3. To counteract the effect of the wind and maintain the plane's southward direction, the pilot needs to point the plane in the opposite direction of the wind's east and north components.
Plane's target heading:
South component = Plane's south component
= 734 km/h
North component = -North component of wind
= -77.78 km/h
East component = -East component of wind
= -77.78 km/h
4. Finally, we can calculate the direction by finding the angle between the southward velocity and the target heading.
Tanθ = North component / East component
Tanθ = (-77.78 km/h) / (-77.78 km/h)
Tanθ = 1
Since Tanθ = 1, the angle θ is 45 degrees.
Therefore, the pilot should aim the plane 45 degrees west of south in order to fly due south when there is a southwest wind blowing at 110 km/h.