An airplane is flying from Montreal to Vancouver. The wind is blowing from the west at 60km/hour. The airplane flies at an airspeed of 750km/h and must stay on a heading of 65 degrees west of north.
A) what heading should the pilot take to compensate for the wind?
B) what is the speed of the airplane relative to the ground?
We are suppose to use the cosine law and I tried to draw a diagram which formed a triangle. But I have no idea how to go on or what to do
If the apparent heading of the plane is (65+θ)°, and the plane's speed is v, then
60+v*sin65° = 750 sin(65+θ)°
750 cos(65+θ)° = v cos65°
v = 695 and θ=1.6°
So, the plane needs to fly at N66.6°W
and the plane's ground speed is 695 km/hr
Not sure how to use the law of cosines for this one.
To solve this problem, we can break down the velocities of the airplane and the wind into their northward (up) and westward (side) components.
Let's use the following conventions:
- Northward component: Positive when moving north, negative when moving south.
- Westward component: Positive when moving west, negative when moving east.
Given information:
- Airplane airspeed = 750 km/h
- Wind speed = 60 km/h (blowing from the west)
- Heading of the airplane = 65 degrees west of north
Let's solve each part of the problem step by step:
A) What heading should the pilot take to compensate for the wind?
To compensate for the wind, the pilot needs to fly in a direction that offsets the wind's effect. This means the heading of the airplane should be in the opposite direction of the wind.
Step 1: Determine the wind components.
Since the wind is blowing from the west (side), the westward component of the wind is +60 km/h.
Step 2: Determine the airplane's heading components.
Since the airplane's heading is 65 degrees west of north, we can split the airspeed into its northward and westward components.
Finding the northward component:
Northward component = airspeed * cosine(heading angle)
Northward component = 750 km/h * cos(65 degrees)
Finding the westward component:
Westward component = airspeed * sine(heading angle)
Westward component = 750 km/h * sin(65 degrees)
Step 3: Calculate the compensated heading.
To find the compensated heading, we need to subtract the wind's westward component from the airplane's westward component.
Compensated heading = arctan(northward component / (airspeed - westward component))
Plug in the values we have:
Compensated heading = arctan((airspeed * cosine(heading angle)) / (airspeed - wind westward component))
Compensated heading = arctan((750 km/h * cos(65 degrees)) / (750 km/h - 60 km/h))
Solve this equation using a calculator or trigonometric table, and you will get the compensated heading.
B) What is the speed of the airplane relative to the ground?
To find the speed of the airplane relative to the ground, we need to calculate the magnitude of the resultant velocity vector.
Step 1: Calculate the northward component of the airplane's resultant velocity.
Northward component = northward component of the airplane's heading + northward component of the wind
Northward component = airspeed * cosine(heading angle) + 0 (since the wind does not blow in the northward direction)
Northward component = airspeed * cosine(heading angle)
Step 2: Calculate the westward component of the airplane's resultant velocity.
Westward component = westward component of the airplane's heading + westward component of the wind
Westward component = airspeed * sine(heading angle) + wind westward component
Westward component = airspeed * sine(heading angle) + 60 km/h
Step 3: Calculate the magnitude (speed) of the airplane's resultant velocity.
Speed = sqrt(northward component^2 + westward component^2)
Plug in the values we have and solve for the speed using a calculator.
Final note: Make sure to double-check all your values, angles, and calculations to avoid errors.
To solve this problem, we can use the concept of vector addition and the cosine law. Let's break it down step by step:
Step 1: Draw a diagram
Start by drawing a diagram to represent the situation. On your diagram, draw a triangle with a northward direction, representing the airplane's intended path. Label the sides and angles of the triangle.
Here's a simplified representation of the diagram:
/|
/ |
Wind /_____| Plane
|
|
Ground
Step 2: Determine the wind's effect on the airplane's heading
Since the wind is blowing from the west, it will have an effect on the airplane's heading. We need to calculate the angle between the airplane's intended heading and the actual heading it needs to take to compensate for the wind.
In this case, the airplane needs to head 65 degrees west of north. However, because of the wind, the airplane will need to adjust its heading.
To calculate the adjusted heading, we need to find the angle between the north direction and the plane's intended direction of 65 degrees west of north. This angle can be found by subtracting 65 degrees from 90 degrees (since the angle between north and west is 90 degrees).
Adjusted heading angle = 90° - 65° = 25°
Step 3: Use vector addition to find the airplane's relative velocity
The airplane's velocity is the sum of its airspeed and the wind's velocity. To calculate the relative velocity of the airplane, we need to find the vector sum of the airspeed vector and the wind vector.
Given:
Airspeed: 750 km/h
Wind: 60 km/h
To find the relative velocity (ground speed), we need to find the magnitude and direction of the resultant vector.
Step 4: Apply the cosine law to find the resultant vector's magnitude
Using the cosine law, we can find the magnitude of the resultant vector:
Resultant magnitude = Square root of [(Airspeed)^2 + (Wind)^2 - 2 × Airspeed × Wind × cos(Adjusted heading angle)]
Plug in the values:
Resultant magnitude = Square root of [(750)^2 + (60)^2 - 2 × 750 × 60 × cos(25°)]
Now, calculate the value to get the resultant magnitude.
Step 5: Calculate the airplane's ground speed
The airplane's ground speed is the magnitude of the resultant vector.
Now that we have the result of Step 4, we can calculate the ground speed by finding the magnitude of the resultant vector:
Ground speed = Resultant magnitude
Step 6: Find the angle between the resultant vector and the north direction
To determine the direction of the airplane's motion relative to the ground, we need to find the angle between the resultant vector and the north direction.
You can use the sine law or trigonometry to find this angle.
With these steps, you should be able to answer both parts A) and B) of the question!
I don't know what your diagram looks like, but try this
I will assume that this is a question dealing with vectors.
let O be the centre of you NS, EW grid
draw a line OB with a heading of N65°W , and mark it x km/h
this will be your resultant vector
draw in your wind vector AB as a horizontal line and mark it 60 km/h
Join AO and mark it 750 km/h
In triangle OBA, angle ABO = 155°
let angle AOB = Ø
by the sine law,
sinØ/60 = sin155/750
Ø = 1.94°
so he should head in the direction of N 66.94° W
then angle A = 23.06°
x/sin23.06 = 750/sin155
x = 695.19 km/h
using the cosine law would be more difficult,
e.g.
750^2 = x^2 + 60^2 - 2(x)(60)cos155
558900 = x^2 + 108.7569..x
x^2 + 108.7569x - 558900 = 0
using the quadratic formula ...
x = 695.19 as above or x = a negative, which must be rejected.
Then you would use the sine law to find the angle