An airplane with a speed of 485 km/h is traveling with a heading of 83.5°. The wind velocity is 38 km/h in the direction of 191°. Find the ground speed of the airplane.
P.S. How do you solve this problem by only using the law of cosine/sine method?
Draw the velocity vector of the plane
add the vector for the wind
The resultant s is the ground speed.
The angle between the two vectors is 72.5°
s^2 = 485^2 + 38^2 - 2*485*38*cos72.5°
To solve this problem using the law of cosines and law of sines method, we can break down the velocities into their components.
Let's start by finding the x and y components of the airplane velocity. The x-component (horizontal component) is given by:
Vx_airplane = airplane_speed * cos(heading)
Vx_airplane = 485 km/h * cos(83.5°)
Next, we find the y-component (vertical component) of the airplane velocity:
Vy_airplane = airplane_speed * sin(heading)
Vy_airplane = 485 km/h * sin(83.5°)
Next, let's find the x and y components of the wind velocity. The x-component is given by:
Vx_wind = wind_speed * cos(wind_direction)
Vx_wind = 38 km/h * cos(191°)
The y-component of the wind velocity is given by:
Vy_wind = wind_speed * sin(wind_direction)
Vy_wind = 38 km/h * sin(191°)
Now, we can find the ground speed of the airplane. The x-component of the ground speed is the sum of the airplane's x-component and the wind's x-component:
Vx_ground = Vx_airplane + Vx_wind
The y-component of the ground speed is the sum of the airplane's y-component and the wind's y-component:
Vy_ground = Vy_airplane + Vy_wind
Using the Pythagorean theorem, we can find the magnitude of the ground speed:
Ground_speed = sqrt(Vx_ground^2 + Vy_ground^2)
To find the numerical value of the ground speed, you can substitute the given values into the equations and solve it using a calculator or math software.