A pilot wishes to fly 450 km due south in 3 hours. A wind is blowing from the west at 50 km/hr. By means of a vector diagram, compute the proper heading and speed that the pilot must choose to achieve this objective.
From the origin,draw a vector on the positive x axis pointing eastward.Label
it 50 km/h.
Draw a vector on the negative y axis pointing south. Label it -150 km/h.
From the origin, draw the resultant vector between the two original vectors
and pointing southeast.
tanA = Y/X = -150/50 = -3,
A = -71.6 deg. = 288.4 deg. CCW.
R = X / cosA = 50 / cos288.4 = 158.4km / h .
288.4 - 270 = 18.4 deg East of the
intended course.
To allow for windage, the plane must head 18.4 deg west of due South:
270 - 18.4 = 25i.6 deg.CCW.
d = 3 h * 158.4 km/h = 475.2 km.
To determine the proper heading and speed for the pilot in order to achieve their objective of flying 450 km due south in 3 hours against a westward wind of 50 km/hr, we can use vector subtraction.
Step 1: Draw a diagram
Draw a diagram representing the situation.
- Draw a horizontal line to represent the ground, labeling it with an arrow pointing to the right (east) to indicate the wind blowing from the west.
- Draw a vertical line beneath the horizontal line to represent the desired direction of travel for the pilot, labeling it with an arrow pointing downward to indicate flying due south.
Step 2: Calculate the wind's effect
From the diagram, the length of the horizontal line represents the wind speed, which is 50 km/hr.
- Choose a suitable scale for the diagram, such as 1 cm = 10 km/hr.
- Draw a line segment of length 5 cm (50 km/hr / 10 km/hr per 1 cm), starting from the right end of the horizontal line, in the opposite direction of the wind (to the left).
Step 3: Calculate the desired airspeed
The desired airspeed is equal to the groundspeed minus the wind speed.
- Since the pilot wishes to fly due south, the desired groundspeed is the same as the desired airspeed.
- To ensure the aircraft moves south at the desired pace of 450 km in 3 hours, the airspeed needs to be 150 km/hr (450 km / 3 hours).
Step 4: Draw the desired airspeed vector
Draw a line segment of length 15 cm (150 km/hr / 10 km/hr per 1 cm) from the end of the horizontal line opposite the wind's effect.
Step 5: Connect the wind-corrected heading
From the starting point of the wind's effect vector, draw a line segment connecting to the end of the desired airspeed vector. This line represents the proper heading the pilot must choose.
- Measure the angle between the horizontal line (representing the ground) and the line segment connecting the starting point of the wind's effect vector to the end of the desired airspeed vector. This angle represents the proper heading for the pilot to achieve their objective.
Step 6: Determine the heading and speed
Based on the diagram, the proper heading for the pilot to choose is southeast, and the speed is 150 km/hr (desired airspeed).
Thus, the pilot should choose a heading of southeast and a speed of 150 km/hr to achieve their objective.
To compute the proper heading and speed the pilot must choose, we will use vector addition. Let's break down the problem into two components: the pilot's desired heading and the wind's effect on the plane's motion.
1. Determine the desired heading (direction):
The pilot wishes to fly due south, which means the desired heading is 180 degrees or in the direction of a straight line downwards on a map.
2. Find the wind's effect on the plane's motion:
The wind is blowing from the west at 50 km/hr, which means it will create resistance (or drag) on the plane. This resistance will affect the plane's speed and direction.
To find the effect of the wind on the plane, we need to determine the wind's component in the north-south (vertical) direction.
The wind is coming from the west, which is at 270 degrees on a compass. To find the north-south component of the wind, we need to use trigonometry.
The north-south component of the wind can be calculated using:
sin(270) = (north-south component of the wind) / (wind's magnitude)
Simplifying the equation:
sin(270) = (north-south component of the wind) / 50
Since sin(270) equals -1, we can solve for the north-south component of the wind:
-1 = (north-south component of the wind) / 50
Therefore, the north-south component of the wind is -50 km/hr.
3. Compute the resultant velocity vector:
To find the proper heading and speed the pilot must choose, we need to compute the resultant velocity vector by adding the desired velocity vector (450 km/hr due south) and the wind's velocity vector (-50 km/hr due north).
We can write the resultant velocity vector as: v_resultant = v_pilot + v_wind
In terms of components, we can write this as:
v_resultant = (v_pilot_north + v_wind_north) î + (v_pilot_south + v_wind_south) ĵ
Since the pilot wishes to fly due south, the north component of the pilot's velocity is 0 km/hr (v_pilot_north = 0), and the south component is 450 km/hr (v_pilot_south = 450 km/hr).
Substituting the values:
v_resultant = (0 + 0) î + (450 + (-50)) ĵ
v_resultant = 0 î + 400 ĵ
Therefore, the resultant velocity vector is 400 km/hr due south.
4. Determine the proper heading and speed:
The proper heading is the direction of the resultant velocity vector. In this case, the proper heading is due south (180 degrees).
The proper speed is the magnitude of the resultant velocity vector, which is 400 km/hr.
To achieve the objective of flying 450 km due south in 3 hours with a westerly wind of 50 km/hr, the pilot must choose a heading of 180 degrees (due south) and a speed of 400 km/hr.