An aircraft have a speed in still air of 400 km/h. A wind is blowing from the south at 80 km/h.
The pilot wishes the aircraft to travel due east. Find the direction in which aircraft should be steered and the speed at which it will travel.
for every hour, the aircraft goes 80km. To go south to compensate, the 400SinTheta=80 on speed, or theta=.2 or Theta will be arc sin .2 S of East.
To find the direction and speed at which the aircraft should be steered to travel due east, we need to consider the effect of the wind on the aircraft's motion.
The aircraft's speed in still air is 400 km/h. However, because there is a wind blowing from the south at 80 km/h, the actual speed of the aircraft relative to the ground will be affected.
Let's break down the velocity vectors:
1. The velocity of the aircraft in still air (Vaircraft) = 400 km/h in the east direction.
2. The velocity of the wind (Vwind) = 80 km/h in the south direction.
Now, to find the resulting velocity of the aircraft relative to the ground, we need to add the velocities vectorially. This can be done using vector addition.
First, draw a "vector diagram" with the velocity of the aircraft in still air as one vector (pointing east) and the velocity of the wind as another vector (pointing south). The resulting velocity vector will be the diagonal of the parallelogram formed by these two vectors.
To find the magnitude and direction of the resulting vector, we can use the Pythagorean theorem and trigonometry. The magnitude of the resulting vector represents the speed of the aircraft relative to the ground, and the direction represents the heading of the aircraft.
Let's calculate the magnitude and direction:
Magnitude:
The magnitude of the resulting vector can be found using the Pythagorean theorem:
Resultant magnitude = sqrt((Vaircraft)^2 + (Vwind)^2)
Substituting the values, we get:
Resultant magnitude = sqrt((400 km/h)^2 + (80 km/h)^2) = sqrt(160000 km^2/h^2 + 6400 km^2/h^2) = sqrt(166400 km^2/h^2) ≈ 408 km/h
Direction:
The direction of the resulting vector can be found using trigonometry. We can use the tangent function to determine the angle between the resulting vector and the east direction.
Tangent of the angle = (Vwind / Vaircraft)
Angle = arctan(Vwind / Vaircraft)
Substituting the values, we get:
Angle = arctan(80 km/h / 400 km/h) = arctan(0.2)
Using a calculator or reference, we find:
Angle ≈ 11.31 degrees
Therefore, the aircraft should be steered at approximately 11.31 degrees north of east, and it will travel at a speed of approximately 408 km/h.