an airplane has a speed of 600km/h with respect to air. it has to fly a distance of 800km northward. in what direction should it fly if a steady wind is blowing at a speed of 120km/h from the west ? what will its speed with respect to the ground be? draw the relevant diagram for your calculation
To determine the direction that the airplane should fly, we need to consider both the speed of the airplane and the speed of the wind.
First, let's draw a diagram to visualize the scenario:
```
120 km/h
Wind ---------->
_________________________
| |
| | *
| | / |
| | / |
| |/ | 800 km
| | |
|<--- 600 km/h --->| |
|_______________________|
```
In the diagram, we have represented the airplane's speed of 600 km/h with respect to the air as an arrow pointing northwards. The wind speed of 120 km/h is represented as an arrow pointing from the west to the east.
To find the direction that the airplane should fly, we need to consider the relative motion between the airplane and the wind. The resultant velocity will be the vector sum of the airplane's velocity and the wind's velocity.
Using vector addition, we can determine the resultant velocity:
```
120 km/h
Wind ---------->
_________________________
| |
| R | *
| /| | / |
| / | | / |
| / | |/ | 800 km
| / | | |
|<--- 600 km/h --->| |
|_______________________|
```
The resultant velocity of the airplane, represented by vector R, will be the diagonal of the parallelogram formed by the airplane's velocity vector and the wind's velocity vector.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
R^2 = (600 km/h)^2 + (120 km/h)^2
R^2 = 360,000 km^2/h^2 + 14,400 km^2/h^2
R^2 = 374,400 km^2/h^2
R = √(374,400) km/h
R ≈ 612.37 km/h
Now that we know the magnitude of the resultant velocity, we can find its direction by calculating the angle it makes with the northward direction.
tan(θ) = opposite/adjacent
tan(θ) = 120 km/h / 600 km/h
θ = arctan(0.2)
θ ≈ 11.31°
Therefore, the airplane should fly at an angle of approximately 11.31° east of north.