An airplane has a speed 590 km/h . If a wind begins blowing from the southwest at a speed of 94.9 km/h (average). In what direction should the pilot aim the plane so that it will fly due south?
you want to fly west to counteract the wind trying to blow you east.
ArcTan(94.9/590) W of S might be a good course. Draw the diagram and see.
To determine the direction the pilot should aim the plane, we need to consider the effect of the wind on the plane's motion. Since the wind is blowing from the southwest, it will create a force against the plane, tending to push it in a northeasterly direction.
To find the direction the pilot should aim the plane, we can use a technique called vector addition. We'll treat the plane's velocity and the wind's velocity as vectors.
First, let's break down the velocities into their north-south and east-west components. We'll assume that the positive direction of north is directly upwards.
The plane's velocity: 590 km/h due south. Since it is going due south, its north-south component is -590 km/h (negative because it's in the opposite direction) and its east-west component is 0 km/h.
The wind's velocity: 94.9 km/h southwest. To break it down into components, we need to use trigonometry. The southwest direction is 45 degrees between south and west.
So, the wind's north-south component is sin(45) * 94.9 km/h and its east-west component is cos(45) * 94.9 km/h.
Calculating these values, we get:
Wind's north-south component = sin(45) * 94.9 = 67.1 km/h (rounded to one decimal place)
Wind's east-west component = cos(45) * 94.9 = 67.1 km/h (rounded to one decimal place)
Now, let's add the plane's and wind's vectors together to find the resulting velocity:
Resultant north-south component = plane's north-south component + wind's north-south component
Resultant north-south component = -590 km/h + 67.1 km/h = -522.9 km/h
Resultant east-west component = plane's east-west component + wind's east-west component
Resultant east-west component = 0 km/h + 67.1 km/h = 67.1 km/h
So, the resulting velocity, in terms of its components, is -522.9 km/h in the north-south direction and 67.1 km/h in the east-west direction.
To determine the direction the pilot should aim the plane, we can find the angle between the resultant velocity and the south direction using trigonometry:
Direction angle = tan⁻¹(resultant east-west component / resultant north-south component)
Direction angle = tan⁻¹(67.1 km/h / -522.9 km/h) ≈ -7.5 degrees (rounded to one decimal place)
The negative sign indicates that the angle is counter-clockwise from the south direction.
Therefore, the pilot should aim the plane approximately 7.5 degrees to the right (east) of due south to counteract the effect of the wind and maintain a southward flight.