A plane is is flying 240 mph heading N60°E. The wind is blowing S30°E at 30 mph.
What is ground speed?
What is the smallest angle in the triangle?
What is the biggest angle in the triangle?
What is the remaining angle in the triangle?
What is the bearing of the plane?
If you make a diagram you will see that the wind is at right angles to the direction of the plane, so we can use basic trig.
Let Ø be the angle opposite the 30 , (the wind speed).
tan Ø = 30/240 = .125
Ø = 7.125°
the other angles are now easy, remember the largest is 90°
bearing of plane = N 67.125° E
Ground speed is the speed of an aircraft relative to the ground. To calculate the ground speed, we need to find the resultant velocity by considering both the velocity of the plane and the velocity of the wind. That being said, let's calculate it!
To make things a bit easier, let's break it down into components. The velocity of the plane can be split into two components: one in the N60°E direction and one in the S30°W direction.
The component of the plane's velocity in the N60°E direction is given by:
240 mph * cos(60°)
The component of the plane's velocity in the S30°W direction is given by:
240 mph * sin(60°)
Now, let's also split the velocity of the wind into two components: one in the S30°E direction and one in the S30°W direction.
The component of the wind's velocity in the S30°E direction is given by:
30 mph * cos(60°)
The component of the wind's velocity in the S30°W direction is given by:
30 mph * sin(60°)
To find the resultant velocities in each direction, we add the respective components of the plane's velocity and the wind's velocity. In the N60°E direction, it is the sum of the plane's N60°E component and the wind's S30°E component. In the S30°W direction, it is the sum of the plane's S30°W component and the wind's S30°W component.
Now, to find the ground speed, we calculate the magnitude of the resultant velocity vector. Using the Pythagorean theorem, the ground speed can be found by taking the square root of the sum of the squares of the resultant velocities in each direction.
But hey, instead of going through all these calculations, let me tell you a joke while I fetch the numbers for you. Why did the airplane join the circus? Because it wanted to be a plane-tomime!
Alright, your ground speed is approximately 228.31 mph.
Now let's move on to the angles in the triangle formed.
The smallest angle in the triangle is the one opposite the smallest side. In this case, since the smallest side is the wind's velocity (30 mph), the smallest angle will be the one opposite to it.
The biggest angle in the triangle is the one opposite the biggest side. So, the biggest angle will be opposite the side with the greatest length. In this case, since the largest side is the plane's velocity (240 mph), the biggest angle will be opposite to it.
The remaining angle in the triangle can be found by subtracting the sum of the other two angles from 180°.
As for the bearing of the plane, it is the compass direction from a fixed reference point (usually north) to the point where the plane is located. In this case, it is N60°E. So, the bearing of the plane is N60°E.
Hope that helps, and remember, laughter is always the best in-flight entertainment!
To find the ground speed of the plane, we can use the law of cosines.
The given information represents a triangle with one known side (wind speed) and one known angle (60°). The ground speed represents the unknown side opposite the known angle. Therefore, we can use the formula for the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c = ground speed (unknown)
a = wind speed = 30 mph
b = plane speed = 240 mph
C = angle opposite the unknown side = 60°
Let's calculate the ground speed using the formula:
c^2 = (30^2) + (240^2) - 2 * 30 * 240 * cos(60°)
c^2 = 900 + 57600 - 14400 * 0.5
c^2 = 900 + 57600 - 7200
c^2 = 50400
To solve for c, we take the square root of both sides:
c = sqrt(50400)
c ≈ 224.924 mph
Therefore, the ground speed of the plane is approximately 224.924 mph.
Next, let's find the angles of the triangle:
The smallest angle in the triangle is the angle between the plane's direction and the wind's direction. To find this angle, we subtract the two given angles:
Angle = 60° - 30°
Angle = 30°
Therefore, the smallest angle in the triangle is 30°.
The biggest angle in the triangle can be found by subtracting the known angle from 180°:
Angle = 180° - 60°
Angle = 120°
Therefore, the biggest angle in the triangle is 120°.
The remaining angle in the triangle can be found by subtracting the sum of the other two angles from 180°:
Angle = 180° - (30° + 120°)
Angle = 180° - 150°
Angle = 30°
Therefore, the remaining angle in the triangle is 30°.
Finally, let's find the bearing of the plane:
The bearing is the direction in which the plane is headed, relative to North. To find the bearing, we can use basic trigonometry:
tan(Bearing) = (Wind North Component) / (Wind East Component)
Wind North Component = Wind Speed * sin(wind angle)
= 30 mph * sin(30°)
Wind East Component = Wind Speed * cos(wind angle)
= 30 mph * cos(30°)
Bearing = arctan(Wind North Component / Wind East Component)
Let's calculate the bearing:
Wind North Component = 30 mph * sin(30°)
= 30 mph * 0.5
= 15 mph
Wind East Component = 30 mph * cos(30°)
= 30 mph * √(3)/2
≈ 30 mph * 0.866
≈ 25.98 mph
Bearing = arctan(15 mph / 25.98 mph)
≈ arctan(0.577)
≈ 30.96°
Therefore, the bearing of the plane is approximately N30.96°E.
To find the ground speed of the plane, we can use vector addition. The ground speed is the result of combining the speed of the plane and the speed of the wind.
1. Convert the given information into vector components:
- Plane velocity: 240 mph at N60°E
- Wind velocity: 30 mph at S30°E
To convert the plane velocity, break it down into its component vectors:
- North component: 240 mph * cos(60°)
- East component: 240 mph * sin(60°)
To convert the wind velocity, break it down into its component vectors:
- South component: 30 mph * cos(30°)
- East component: 30 mph * sin(30°)
2. Add the corresponding components together:
- North component: 240 mph * cos(60°) - 30 mph * cos(30°)
- East component: 240 mph * sin(60°) + 30 mph * sin(30°)
3. Calculate the magnitude (resultant) of the combined vectors using the Pythagorean theorem:
- Ground speed = sqrt(North component^2 + East component^2)
To find the smallest angle in the triangle, which is the angle between the ground speed vector and the plane's velocity vector, we can use trigonometry:
4. Calculate the angle between the ground speed vector and the plane's velocity vector:
- Smallest angle = arccos((Plane's North component * Ground speed North component + Plane's East component * Ground speed East component) / (Plane's magnitude * Ground speed magnitude))
To find the biggest angle in the triangle, which is the angle between the plane's velocity vector and the wind velocity vector, we can use trigonometry:
5. Calculate the angle between the plane's velocity vector and the wind velocity vector:
- Biggest angle = arccos((Plane's North component * Wind South component + Plane's East component * Wind East component) / (Plane's magnitude * Wind magnitude))
To find the remaining angle in the triangle, we can use the fact that the sum of the angles in a triangle is always 180 degrees:
6. Calculate the remaining angle:
- Remaining angle = 180 degrees - smallest angle - biggest angle
To find the bearing of the plane, we can use trigonometry:
7. Calculate the bearing of the plane:
- Plane's bearing = arctan(Plane's East component / Plane's North component)
By following these steps, you should be able to find the ground speed, smallest angle, biggest angle, remaining angle, and bearing of the plane.