My Question is :

A plane with an air pf 192 miles per hour is headed on a bearing of 121 degrees. A north wind is blowing ( from north to south ) at 15.9 miles per hour. Find the ground speed and the actual bearing of the plane.

heading 121 - 90 = 31 degrees south of east

South speed = 192 sin 31 + 15.9 = 114.8
East speed = 192 cos 31 = 164.5
tan of angle S of E = 114.8/164.5 = .6979
angle S of E = 34.9
so course made good is 90+34.9 =124.9 degrees

V = 192[121o] + 15.9[180o]. mi/h.

X = 192*sin121 + 15.9*sin180 = 164.6. mi/h.
Y = 192*Cos121 + 15.9*Cos180 = -114.8 mi/h.

V = 164.6 - 114.8i = 200.7mi/h[-55.1o] = 200.7mi/h[55.1o] S. of E. = 200.7mi/h[145.1o] CW.
Tan A = X/Y

To find the ground speed and the actual bearing of the plane, we can use vector addition.

Let's denote the airspeed of the plane as V and the wind speed as W. The ground speed of the plane is the vector sum of the airspeed and wind speed.

We can break down the airspeed V into its northward component and eastward component using trigonometry. The northward component, denoted as V_n, is given by V * sin(theta), where theta is the bearing angle. Similarly, the eastward component, denoted as V_e, is given by V * cos(theta).

Given:
V = 192 mph (airspeed)
theta = 121 degrees (bearing angle)
W = 15.9 mph (wind speed)

First, calculate the northward component of the airspeed using the sine function:
V_n = V * sin(theta)
= 192 * sin(121 degrees)

Next, calculate the eastward component of the airspeed using the cosine function:
V_e = V * cos(theta)
= 192 * cos(121 degrees)

Now, let's add the wind speed vector to the airspeed vector to find the resultant velocity vector, which represents the ground speed and direction of the plane.

The northward component of the resultant velocity, denoted as R_n, is the sum of the northward components of the airspeed and wind speed:
R_n = V_n + W

The eastward component of the resultant velocity, denoted as R_e, is the sum of the eastward components of the airspeed and wind speed:
R_e = V_e

Finally, we can find the magnitude (ground speed) and direction (bearing) of the resultant velocity using the Pythagorean theorem and inverse tangent function:

The magnitude (ground speed), denoted as R, is given by:
R = sqrt(R_n^2 + R_e^2)

The direction (bearing) of the resultant velocity, denoted as phi, is given by:
phi = arctan(R_e / R_n)

Now, let's calculate the values:

V_n = 192 * sin(121 degrees) ≈ 73.47 mph
V_e = 192 * cos(121 degrees) ≈ -154.12 mph (negative because it's pointing westward)

R_n = V_n + W ≈ 73.47 mph + 15.9 mph ≈ 89.37 mph
R_e = V_e ≈ -154.12 mph

R = sqrt(R_n^2 + R_e^2) ≈ sqrt((89.37 mph)^2 + (-154.12 mph)^2) ≈ 178.98 mph
phi = arctan(R_e / R_n) ≈ arctan((-154.12 mph) / (89.37 mph)) ≈ -59.46 degrees

Therefore, the ground speed of the plane is approximately 178.98 mph, and the actual bearing is approximately 300.54 degrees (calculated by adding 180 degrees to the angle -59.46 degrees).

To find the ground speed (the speed of the plane as observed from the ground) and the actual bearing (the direction in which the plane is traveling), we can use vector addition.

Start by breaking down the motion of the plane into two components: the east-west component and the north-south component.

1. East-West Component:
To find the east-west component, we can use trigonometry. The east-west component of the plane's airspeed is given by:
Speed = Airspeed * cos(Bearing)

Given:
Airspeed = 192 miles per hour
Bearing = 121 degrees

Substituting the values into the formula, we have:
East-West Component = 192 * cos(121 degrees)

2. North-South Component:
Similarly, the north-south component of the plane's airspeed is given by:
Speed = Airspeed * sin(Bearing)

Given:
Airspeed = 192 miles per hour
Bearing = 121 degrees

Substituting the values into the formula, we have:
North-South Component = 192 * sin(121 degrees)

3. Ground Speed:
To find the ground speed, we need to take into account the wind speed as well. The ground speed is the resultant of the plane's airspeed and the wind speed. We can add the east-west components and north-south components separately to get the total east-west speed and north-south speed.

Total East-West Speed = East-West Component of Airspeed + East-West Component of Wind Speed
Total North-South Speed = North-South Component of Airspeed + North-South Component of Wind Speed

Given:
Wind Speed = 15.9 miles per hour

Substituting the values into the formulas, we have:
Total East-West Speed = (192 * cos(121 degrees)) + (15.9 * cos(180 degrees))
Total North-South Speed = (192 * sin(121 degrees))- (15.9 * sin(180 degrees))

The ground speed can be calculated by finding the magnitude of the resultant vector of the east-west and north-south speeds:
Ground Speed = sqrt((Total East-West Speed)^2 + (Total North-South Speed)^2)

4. Actual Bearing:
The actual bearing of the plane can be found using the inverse tangent function:
Bearing = atan(Total North-South Speed / Total East-West Speed)

Now, you can substitute the values into the formulas and perform the calculations to find the ground speed and the actual bearing of the plane.