An airplane leaves Skyharbor airport at 8:30 PM. His average flight speed is 585 mph. He travels at 43 degrees east of north. The first half of his flight he encounters a cross wind of 72 mph at a direction of 70 degrees west of north. The second half of the flight he encounters a tailwind of 53 mph which hits the plane at 55 degrees west of south. He lands at his destination in 4 hours and 56 minutes. What is his final position vector from Skyharbor airport?
Write the vector in component form
What is his final distance from skyharbor
What is his final direction or bearing from skyharbor
Let's do the various legs in component form:
585@N43°E = <399,428>
72@N70°W = <-68,25>
53@S55°W = <-43,-30>
To get the various distances, involved, multiply these speeds by the time spent flying. Then you can convert the final displacement back to polar form.
does it not matter if the 585 is the average speed?
only if you're interested in speed at various places along the way. But you are ultimately only interested in the distance traveled, which is avg speed * time
To find the final position vector from Skyharbor airport, we can use vector addition.
First, let's calculate the horizontal and vertical components of the airplane's velocity during the first half of the flight.
Horizontal Component:
The airplane travels at 585 mph in the direction 43 degrees east of north.
Since the crosswind is at 70 degrees west of north, we need to subtract its horizontal component from the airplane's velocity.
The horizontal component of the crosswind is 72 mph * cos(70 degrees).
Therefore, the horizontal component of the airplane's velocity during the first half of the flight is:
585 mph * sin(43 degrees) - 72 mph * cos(70 degrees)
Vertical Component:
The airplane travels at 585 mph in the direction 43 degrees east of north.
Since the crosswind is at 70 degrees west of north, we need to subtract its vertical component from the airplane's velocity.
The vertical component of the crosswind is 72 mph * sin(70 degrees).
Therefore, the vertical component of the airplane's velocity during the first half of the flight is:
585 mph * cos(43 degrees) - 72 mph * sin(70 degrees)
Next, let's calculate the horizontal and vertical components of the airplane's velocity during the second half of the flight.
Horizontal Component:
The tailwind hits the plane at 55 degrees west of south, which is equivalent to 125 degrees counter-clockwise from the positive x-axis.
The horizontal component of the tailwind is 53 mph * sin(125 degrees).
Therefore, the horizontal component of the airplane's velocity during the second half of the flight is:
53 mph * sin(125 degrees)
Vertical Component:
The tailwind hits the plane at 55 degrees west of south, which is equivalent to 125 degrees counter-clockwise from the positive x-axis.
The vertical component of the tailwind is 53 mph * cos(125 degrees).
Therefore, the vertical component of the airplane's velocity during the second half of the flight is:
53 mph * cos(125 degrees)
Now, we can calculate the total horizontal and vertical components of the airplane's velocity.
Total Horizontal Component:
Add the horizontal components of the first and second half of the flight together:
(585 mph * sin(43 degrees) - 72 mph * cos(70 degrees)) + (53 mph * sin(125 degrees))
Total Vertical Component:
Add the vertical components of the first and second half of the flight together:
(585 mph * cos(43 degrees) - 72 mph * sin(70 degrees)) + (53 mph * cos(125 degrees))
To find the final position vector, multiply the total horizontal and vertical components by the duration of the flight, which is 4 hours and 56 minutes (or 4.93333 hours).
Final Position Vector:
(4.93333 hours) * (Total Horizontal Component) = x-component of the final position vector
(4.93333 hours) * (Total Vertical Component) = y-component of the final position vector
Once you calculate the x and y components, you can write the final position vector in component form as <x, y>.
To find the final distance from Skyharbor, use the Pythagorean theorem:
Final Distance = sqrt((x-component)^2 + (y-component)^2)
To find the final direction or bearing from Skyharbor, use trigonometry:
Final Direction = arctan(y-component / x-component)