A pilot maintains a heading due west with an air speed of 24o km/h. After flying for 30 min, he finds himself over a town that he knows is 150 km west and 40 km south of his starting point.
a) Waht is the wind velocit, in magniture and direction?
b) What heading should he now maintain, with the same air speed, to follow a course due west from the town?
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To calculate the wind velocity and direction, we can use the fact that the pilot flew west at an airspeed of 240 km/h for 30 minutes. We can assume that the wind blew along a straight path from the starting point to the town, making it a direct crosswind.
a) To find the wind velocity, we need to calculate the ground speed of the plane. Ground speed is the combination of the airspeed and the wind velocity. We can use the Pythagorean theorem to find the ground speed.
First, let's convert the flight time from minutes to hours:
30 minutes = 30/60 = 0.5 hours
Now, we can use the formula for speed (distance divided by time):
Ground Speed = Distance / Time
The distance traveled by the plane in 0.5 hours is:
Distance = Airspeed * Time
= 240 km/h * 0.5 hours
= 120 km
Given that the town is 150 km west and 40 km south of the starting point, we can use these distances to form a right triangle:
|\
| \
| \ (40 km)
| \
|____\
150 km
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the hypotenuse of the triangle, which represents the ground distance traveled by the plane:
c^2 = 150^2 + 40^2
c = √(150^2 + 40^2)
c ≈ 155.56 km
Now we have the ground distance and the time, so we can find the ground speed:
Ground Speed = Distance / Time
Ground Speed = 155.56 km / 0.5 hours
Ground Speed ≈ 311.12 km/h
To determine the wind velocity and direction, we can subtract the airspeed from the ground speed. Since the airspeed was due west, the wind will be blowing from the west as well.
Wind Speed = Ground Speed - Air Speed
Wind Speed = 311.12 km/h - 240 km/h
Wind Speed ≈ 71.12 km/h (magnitude)
Therefore, the wind velocity is approximately 71.12 km/h, blowing from the west.
b) To follow a course due west from the town, the pilot should maintain a heading that compensates for the wind. Since the wind is blowing from the west, the pilot should adjust the heading to the east to counteract the wind's effect.
The heading can be calculated using trigonometric functions. We can use the cosine function to find the angle between the heading and the direction east (which is 90 degrees).
Cosine = Adjacent / Hypotenuse
Cosine θ = 240 km/h / 311.12 km/h
Now, we can find the angle θ by taking the inverse cosine (arc cosine) of the value obtained:
θ = Cos^(-1)(240 km/h / 311.12 km/h)
θ ≈ 36.29 degrees
Therefore, the pilot should maintain a heading of approximately 36.29 degrees east of due west to follow a course due west from the town.