SOLUTION: An airplane is traveling at a speed of 200 miles/hour with a bearing of 240°. The wind velocity is 60 miles/hour at a bearing of 25°. What are the plane's actual speed and direct

Question 1172269: An airplane is traveling at a speed of 200 miles/hour with a bearing of 240°. The wind velocity is 60 miles/hour at a bearing of 25°. What are the plane's actual speed and direction angle?
a. The speed is 140.33 miles/hour, and the direction angle is 237.87°.
b. The speed is 140.73 miles/hour, and the direction angle is 197.15°.
c. The speed is 154.73 miles/hour, and the direction angle is 77.85°.
d. The speed is 154.73 miles/hour, and the direction angle is 197.15°.
thanks in advance for helping me
Found 2 solutions by Solver92311, math_tutor2020:
Answer by Solver92311(821) (Show Source): You can put this solution on YOUR website!

You can do this one in your head. The wind is 5 degrees off of the plane's port bow. So the wind is going to subtract most of 60 mph from the plane's speed and push it slightly to the right. (makes it's bearing a tiny bit smaller)

John

My calculator said it, I believe it, that settles it

From
I > Ø

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

With compass bearings, the angle 000° points directly north. As the angle increases, you turn toward the east direction. This means the angle increases as you turn clockwise as this diagram indicates

Image source:
https://gis.stackexchange.com/questions/239374/determining-spatial-sort-order-with-arcpy

The bearing 240 degrees is located in the southwest quadrant, while the bearing 025 degrees is in the northeast quadrant.

-----------------------
When it comes to angles in trig, convention has 0 degrees pointing directly east.
Also, the angle increases as you move counterclockwise.
So we'll have to make a conversion.

Note how the bearing 240 degrees is 30 degrees below the west direction (270).
For angles in trig, the west direction is 180 degrees.
This means we add on 30 to 180 to get 30+180 = 210

So the bearing 240 degrees is the same as the angle 210 degrees when converting to the trig equivalent angle.

Similarly, the bearing 025 degrees converts to 65 degrees

This diagram visually summarizes what I mean

note: The diagram above showing the plane's vector is not accounting for the wind.

We have the bearing 025° be the same as the angle 65°, when we start 0 aimed at the east direction (instead of north)

The bearing 240° is the same as the angle 210°

The bearings are in red while the converted angle measures are in blue

I broke up the two vectors so they each had their own coordinate system. Though we'll combine them later as explained below.

-----------------------

The vector for the aircraft, without accounting for the wind, is
< x, y > = < r*cos(theta), r*sin(theta) >
< x, y > = < 200*cos(210), 200*sin(210) >
p = < 200*cos(210), 200*sin(210) >

Where r = 200 is the plane's speed without the wind affecting it

The wind's vector is
< x, y > = < r*cos(theta), r*sin(theta) >
< x, y > = < 60*cos(65), 60*sin(65) >
w = < 60*cos(65), 60*sin(65) >

-----------------------

Now we add the two vectors:

p+w = < 200*cos(210), 200*sin(210)>+<60*cos(65), 60*sin(65) >
p+w = < 200*cos(210)+60*cos(65), 60*sin(65)+200*sin(210) >
p+w = < -147.847985052446, -45.621532777801 >
f = < -147.847985052446, -45.621532777801 >
Make sure your calculator is in degree mode.

f represents the final vector, which is where the plane is aimed with the wind accounted for

-----------------------

Let f = < a,b >

We need to find the direction and magnitude for vector f

Recall the magnitude is simply the length of the vector.
We use the pythagorean theorem to see that
|f| = sqrt(a^2+b^2)
|f| = sqrt((-147.847985052446)^2+(-45.621532777801)^2)
|f| = 154.726697557546
|f| = 154.73

After accounting for the wind, the plane's speed is roughly 154.73 mph.

Now find the direction angle theta
theta = arctan(b/a)
theta = arctan(-45.621532777801/(-147.847985052446))
theta = 17.1486847735892
theta = 17.15

This theta value points in quadrant 1 (northeast quadrant), but we want to aim for quadrant 3 (southwest quadrant).
This is because both a,b are negative making the point (a, b) be in quadrant 3.
So we need to add 180 to this result

theta+180 = 17.15+180 = 197.15

After accounting for the wind, the plane's direction is roughly 197.15° (which is in the southwest quadrant)

Answer: d. The speed is 154.73 miles/hour, and the direction angle is 197.15°
RELATED QUESTIONS
An airplane is traveling at a speed of 200 miles/hour with a bearing of 240�. The wind... (answered by Alan3354)
An airplane is traveling at a speed of 240 miles/hour with a bearing of 110°. The wind... (answered by Solver92311)
An airplane is headed northeast, bearing 45◦, with an air speed of 500 miles per... (answered by Alan3354)
a boat is traveling at 30 miles per hour at a bearing of 35 degrees. The river current... (answered by Alan3354)
A jet, traveling at an average speed of 420 miles per hour in still air, flew 240 miles... (answered by ptaylor,checkley77)
A car is traveling at a speed of 60 miles per hour. How many hours will it take for the... (answered by Alan3354)
An airplane flying from Los Angeles to New York is traveling at a velocity of 1,200... (answered by Alan3354)
an airplane can travel at 1000 miles with the wind in the same time it could travel with... (answered by lwsshak3)
A plane flies on a bearing of 230o at 350 miles per hour. A wind is blowing from the... (answered by Alan3354)