# Find ground speed and direction of airplane

In summary, the problem involves finding the ground speed and direction of a plane flying on a bearing of 340 degrees at 400mph, while a wind is blowing at a bearing of 320 degrees at 30mph. After converting the bearings to standard positions, the ground speed is calculated to be approximately 428.214mph. To find the direction, the components of the plane's movement and wind are added together and the arctan is taken. However, since inverse trig functions introduce an ambiguity, the answer should be considered in the context of the physical situation. In this case, the direction is determined to be 338.63 degrees when converted back to navigational convention. However, it would have been easier to simply swap the

## Homework Statement

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"An airplane is flying on a bearing of 340 degrees at 400mph. A wind is blowing at a bearing of 320 degrees at 30mph. Find ground speed and direction of the plane."

vx=vcosϕ
vy=vsinϕ

## The Attempt at a Solution

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First, my teacher told us to change from bearings to standard positions, so I did that. For the plane, I have that it is flying at 110 degrees. For the wind, I have that it is blowing at 130 degrees.

Knowing that, I calculated ground speed by doing the following:
(400cos110+30cos130)^2 + (400sin110+30sin130)^2
Then I took the square root of that and rounded to get a speed of about 428.214mph.

The issue, then, is finding the direction of the plane. I calculated the components of the plane's movement to be <-136.808,375.877> and the wind to be <-19.284,22.981>.

I then added them together for a total of <-156.092,398.858>.

I took the arctan of (398.858/-156.092) to try to get the angle, and I got -68.627 degrees. Now, I know I'm supposed to do something to this because it's not supposed to be negative, but I'm not sure where to go from here.

Are my calculations correct up to this point? If so, how do I use what I have now to find direction?

Thank you very much!

For the plane, I have that it is flying at 110 degrees. For the wind, I have that it is blowing at 130 degrees.
How are you calculating these? You seem to be subtracting from 450.
An airplane is flying on a bearing of 340 degrees
This is misleading. The question means heading, not bearing.

haruspex said:
How are you calculating these? You seem to be subtracting from 450.

My teacher said bearings are based around north, meaning that the 340 and 320 degrees mentioned in the problem are 340 and 320 degrees clockwise from north on a compass, which is 90 degrees on the standard unit circle. That means I had to convert them to what she calls "standard" position in order to do this problem.

If a vector is at 340 degrees from north (moving clockwise), then it is 20 degrees away from the north position because 360-340 is 20. I then added this 20 degrees to 90 degrees (because north corresponds to 90 degrees on the unit circle) to get 110 degrees. Similar reasoning got me to the 130 degrees for the wind.

I don't know if this is the accepted way to do it, but it's what my precalc teacher has us do. Is there an easier/better way to find these?

Regarding headings vs bearings, I'm afraid I don't know much about aviation. I'm just going off what the question says

First, my teacher told us to change from bearings to standard positions, so I did that. For the plane, I have that it is flying at 110 degrees. For the wind, I have that it is blowing at 130 degrees.
OK. Subtracting from 90, then adding 360 if necessary to make it positive is correct.
I took the arctan of (398.858/-156.092) to try to get the angle, and I got -68.627 degrees.
You need to be careful with inverse trig functions. They introduce an ambiguity. arctan cannot distinguish between an angle and that angle plus 180 degrees, so it could be 180-68.63. You need to consider the physical situation.
In view of the convention you converted to, the angle calculated will be anticlockwise from E. You should probably convert back to the navigational convention.

haruspex said:
You need to be careful with inverse trig functions. They introduce an ambiguity. arctan cannot distinguish between an angle and that angle plus 180 degrees, so it could be 180-68.63. You need to consider the physical situation.
In view of the convention you converted to, the angle calculated will be anticlockwise from E. You should probably convert back to the navigational convention.
180-68.63 is 111.37 degrees, which is 338.63 degrees for navigation. Is that the answer the problem is looking for and if so, did I need to convert to "standard" convention at all?

which is 338.63 degrees for navigation
Sounds right.
did I need to convert to "standard" convention at all?
No. The mapping is a reflection about the line x=y, so it would have been easier simply to swap the x and y axes.

haruspex said:
Sounds right.

No. The mapping is a reflection about the line x=y, so it would have been easier simply to swap the x and y axes.
Thank you very much for your help!

## 1. How do you find the ground speed of an airplane?

The ground speed of an airplane can be calculated by dividing the distance traveled by the time taken. This can be done using a GPS system or by using the distance and time measurements from the departure and arrival airports.

## 2. What factors affect the ground speed of an airplane?

The ground speed of an airplane can be affected by various factors such as wind speed and direction, altitude, weight of the airplane, and air temperature. These factors can either increase or decrease the ground speed of an airplane.

## 3. How do you determine the direction of an airplane?

The direction of an airplane can be determined by using a compass or a GPS system. The compass will show the direction of the airplane in relation to magnetic north, while the GPS system will show the direction of the airplane in relation to true north.

## 4. Is the ground speed of an airplane the same as its air speed?

No, the ground speed and air speed of an airplane are not the same. Air speed is the speed at which the airplane is moving through the air, while ground speed is the speed at which the airplane is moving over the ground.

## 5. Why is it important to know the ground speed and direction of an airplane?

Knowing the ground speed and direction of an airplane is important for navigation and flight planning. It helps pilots determine the most efficient route to their destination and to adjust their flight plan in case of changing weather conditions.

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