A small aircraft has a flying speed of 200 km/h in still air.

To make a trip from Saskatoon to Regina, a straight line
distance on the map of 250 km, a pilot must first
determine the wind velocity and then calculate what
direction to fly (the 'heading') to compensate for the
wind. (NOTE: All headings are referenced from the
North, measured positive in the clockwise direction, eg.
East is +90 degrees)
Assume Regina has a heading on the map of 150
degrees from Saskatoon, and that the wind is blowing
steady from North to South. The pilot determines that
the aircraft heading must be 140 degrees in order to
end up in Regina.
a) What is the wind speed?
b) How long will it take to make the trip from Saskatoon
to Regina?

To solve this problem, we'll use the concept of vector addition.

Let's start by understanding the given information:
- The aircraft's flying speed in still air is 200 km/h.
- The straight-line distance between Saskatoon and Regina is 250 km.
- The heading from Saskatoon to Regina on the map is 150 degrees.
- The aircraft must have a heading of 140 degrees relative to North to compensate for the wind and accurately reach Regina.
- The wind is blowing from North to South.

a) To find the wind speed, we'll consider the aircraft's ground speed, which is the speed at which the aircraft travels relative to the ground. It is the vector sum of the aircraft's airspeed and the wind velocity.

In this case, the ground speed vector (200 km/h) should be in the direction of the heading (140 degrees). Since the wind is coming from the North (180 degrees), the resultant vector (ground speed vector) should be in the direction of the heading minus the wind direction.

To calculate the wind speed, we need to use vector components. We can represent the ground speed vector as the sum of two component vectors: one in the North-South direction (wind vector) and the other in the heading direction.

Given:
Heading = 140 degrees
Wind Direction = 0 degrees (since the wind is coming from North to South)

The component vector in the heading direction can be calculated using the formula:
Ground Speed * cos(heading - wind direction)

Component Vector in Heading direction = 200 km/h * cos(140 degrees - 0 degrees)
Component Vector in Heading direction = 200 km/h * cos(140 degrees)
Component Vector in Heading direction ≈ -160 km/h (negative sign indicates opposite direction)

Now, we can find the wind speed by subtracting the component vector in the heading direction from the ground speed vector:
Wind Speed = Ground Speed - Component Vector in Heading direction
Wind Speed = 200 km/h - (-160 km/h)
Wind Speed = 200 km/h + 160 km/h
Wind Speed = 360 km/h

Therefore, the wind speed is 360 km/h.

b) To calculate the time it will take to make the trip from Saskatoon to Regina, we need to consider the ground speed (taking into account the wind speed) and the distance between the two cities.

The time can be determined using the formula:
Time = Distance / Ground Speed

Given:
Distance = 250 km
Ground Speed = 200 km/h + Wind Speed = 200 km/h + 360 km/h = 560 km/h

Time = 250 km / 560 km/h
Time ≈ 0.4464 hours or 26.7886 minutes

Therefore, it will take approximately 0.4464 hours or 26.7886 minutes to make the trip from Saskatoon to Regina.