To answer both of these questions, we need to consider the effect of the wind on the airplane's motion. We can break down the motion into two components: the airplane's velocity relative to the air, and the air's velocity relative to the ground.
A) To determine the heading the pilot should take to compensate for the wind, we need to analyze the vectors involved. Let's assume the airplane's heading is measured clockwise relative to due north.
The given heading "south 60 degrees west" can be converted to a bearing by subtracting it from 360 degrees. So, south 60 degrees west becomes 360° - 60° = 300°.
Now, we need to find the angle between the wind direction and the airplane's velocity relative to the air. The wind is blowing from the west, which is 270 degrees. Therefore, the angle between the wind and the airplane's velocity relative to the air is 300° - 270° = 30°.
So, the pilot should take a heading of 30 degrees to compensate for the wind.
B) To find the speed of the airplane relative to the ground, we can use vector addition to combine the airplane's velocity relative to the air and the wind's velocity relative to the ground.
To make the calculation easier, we can break down each velocity into its north/south and east/west components.
The airplane's velocity is 950 km/h directed south 60 degrees west. Breaking it down, we get:
- North/South component: 950 km/h * sin(60°) = 823.22 km/h (south)
- East/West component: 950 km/h * cos(60°) = 475 km/h (west)
The wind's velocity is 90 km/h blowing from the west. So, its east/west component is -90 km/h.
Now, we can add the components of the airplane's velocity relative to the air and the wind's velocity relative to the ground:
- North/South component: 823.22 km/h
- East/West component: 475 km/h + (-90 km/h) = 385 km/h
Using these components, we can apply the Pythagorean theorem to find the total speed:
Total speed = sqrt((823.22 km/h)^2 + (385 km/h)^2)
Total speed ≈ 892 km/h
Therefore, the speed of the airplane relative to the ground is approximately 892 km/h.