The pilot of an aircraft wishes to fly due west in a 46.8 km/h wind blowing toward the south. The speed of the aircraft in the absence of a wind is 204 km/h.
(a) In what direction should the aircraft head?
° north of west
(b) What should its speed relative to the ground be?
km/h
(a) The pilot should head slightly north of west in order to compensate for the southward wind. Let's call this angle θ.
(b) To find the speed relative to the ground, we need to find the resultant velocity by adding the velocity of the aircraft to the velocity of the wind.
Using components, we can break down the wind's velocity into its northward and westward components.
Southward wind component: 46.8 km/h
Northward wind component: 0 km/h
Westward wind component: -46.8 km/h
Now, let's break down the velocity of the aircraft into its northward and westward components.
Northward velocity component of the aircraft: 0 km/h
Westward velocity component of the aircraft: 204 km/h
To find the resultant velocity, we add these components together:
Resultant northward velocity = 0 km/h + 0 km/h = 0 km/h
Resultant westward velocity = -46.8 km/h + 204 km/h = 157.2 km/h
So, the aircraft should head slightly north of west (θ) and its speed relative to the ground should be 157.2 km/h.
To find the direction in which the pilot should steer the aircraft and the speed of the aircraft relative to the ground, we will use vector addition.
Let's break down the given information into vectors:
1. The speed of the aircraft in the absence of wind: 204 km/h.
This can be represented as a vector pointing due west, since the aircraft wants to fly in that direction.
2. The wind speed: 46.8 km/h, blowing toward the south.
This can be represented as a vector pointing due south.
To find the resulting vector, we need to add the vectors for the aircraft's airspeed and the wind speed.
1. Start by drawing a diagram or coordinate system to visualize the vectors. Let's assume up is the north direction, right is the east direction, and the west and south directions are opposite to east and north, respectively.
2. Draw a vector pointing west to represent the aircraft's airspeed of 204 km/h.
3. Draw a vector pointing south to represent the wind speed of 46.8 km/h.
4. Add the vectors tip-to-tail, i.e., align the tail of the wind vector with the tip of the aircraft's airspeed vector.
5. Draw the resultant vector from the tail of the first vector to the tip of the last vector.
6. Measure the angle between the resultant vector and the west direction. This angle is the direction in which the aircraft should steer.
To find the speed of the aircraft relative to the ground, we need to calculate the magnitude of the resultant vector.
1. Use the Pythagorean theorem to calculate the magnitude of the resultant vector (speed of the aircraft relative to the ground).
Magnitude = sqrt(x^2 + y^2)
where x and y represent the horizontal and vertical components of the resultant vector, respectively.
In this case, since the vectors are orthogonal (perpendicular to each other), we can simply use the Pythagorean theorem.
2. Calculate the magnitude using the formula:
Magnitude = sqrt(204^2 + 46.8^2) km/h
Simplifying gives: Magnitude = sqrt(41616 + 2187.84) km/h
Magnitude = sqrt(43803.84) km/h
Magnitude ≈ 209.2 km/h
So, the direction in which the pilot should head is north of west, and the speed of the aircraft relative to the ground should be approximately 209.2 km/h.
a. X = -204 km/h.
Y = -46.8 km/h.
tanA = Y/X = -46.8/-204 = 0.22941.
Ar = 12.9o South of West.
The plane should head 12.9o North of West.
b. A = 180 + Ar = 180 + 12.9 = 192.9o
V = X/cosA = -204/cos192 = 209.3 km/h.
a. Vpa = Vpg + Va = -204 km/h
Vpg - 46.8i = -204
Vpg = -204 + 46.8i
Tan Ar = Y/X = 46.8/-204 = -0.22941
Ar = -12.9o = Reference angle.
A = -12.9 + 180 = 167.1o CCW = 12.9o N.
of W. = Direction.
b. Vpg = X/Cos A = -204/Cos167.1 = 209.3 m/s. = Velocity of plane relative to gnd.