A pilot wants to fly directly north to an airport 500 km away. His speed relative to the air is 100 km/h. Howevere there is a 40 km/h cross wind blowing from west to east. At what angle relative to the south north direction should he fly to arrive at the distant airport? How long will it take to reach the airport?

Vp + Vw = 100i

Vp + 40 = 100i
Vp = -40 + 100i = Velocity of the plane

Tan A = 100/-40 = -2.50
A = 111.8o CCW = 21.8o W of N

d = V*t = 500 km
t = 500/V = 500/100 = 5 h.

To determine the angle the pilot should fly and the time it will take to reach the airport, we can break down the problem into two components: the wind's effect on the plane's motion and the pilot's flight path.

1. Wind's Effect on the Plane's Motion:
- The wind speed is 40 km/h, blowing from the west to the east.
- Let's consider the wind vector as a displacement vector from the west to the east, pointing towards the east.
- Since the airplane's speed relative to the air is 100 km/h, the actual motion of the airplane is a combination of its own motion and the wind's motion.
- We can calculate the resultant vector of the wind and airplane velocities using vector addition.

Applying vector addition, let's find the resultant vector:

Wind vector (40 km/h to the east)
__________________________
|\ /|
| \ / |
| \ / |
Airplane | \ Resultant vector | Airplane
Vector (100 km/h) \ | Vector (100 km/h)
/ \
/ \
/ \
/ \

The resultant vector represents the actual flight path of the plane relative to the ground.

2. Pilot's Flight Path:
- The pilot wants to fly directly north, so we need to find the angle between the resultant vector and the north direction.
- Since the pilot is flying north, the angle of interest is the angle between the resultant vector and the south-north line.

To find this angle, we can use trigonometry. Let's apply the following steps:
- Divide the airplane's speed (100 km/h) by the wind's speed (40 km/h) to get the tangent of the angle.
- Calculate the inverse tangent (arctan) of this value to find the angle in radians or degrees.

3. Time to Reach the Airport:
- To determine the time it takes for the airplane to reach the airport, we can divide the distance (500 km) by the airplane's speed relative to the air (100 km/h).

Now, let's calculate the angle and the time using the given numbers:

Wind's Effect on the Plane's Motion:
- The airplane's speed relative to the air: 100 km/h
- The wind speed: 40 km/h

The resultant vector (actual flight path) can be found by adding the vectors, as mentioned above, and it has both vertical and horizontal components.

Pilot's Flight Path:
- The angle between the resultant vector and the south-north line is given by the inverse tangent (arctan) of (100/40).

Time to Reach the Airport:
- The distance to the airport: 500 km
- The airplane's speed relative to the air: 100 km/h

By calculating the angle and dividing the distance by the speed, you will find the angle the pilot should fly relative to the south-north direction and the time it will take to reach the airport.