An airplane flies at airspeed (relative to the
air) of 250 km/h . The pilot wishes to fly due
North (relative to the ground) but there is a
65 km/h wind blowing Southwest (direction
225◦
).
In what direction should the pilot head the
plane (measured clockwise from North)?
Answer in units of ◦
Please Explain
You are given to vectors of the triangle
Resultant=wind+heading
heading=resulting-wind
break up resultant, and wind, into N, and E components, subtract, and you then have the components of heading. You can figure heading by theta=arctan(East/North).
To determine in what direction the pilot should head the plane, we need to determine the resultant velocity, which is the vector sum of the airplane's airspeed and the wind velocity.
First, let's break down the airplane's airspeed and the wind velocity into their horizontal and vertical components.
The airspeed of the airplane is given as 250 km/h. Since the pilot wants to fly due North, the horizontal component of the airspeed will be zero (as the airplane is not moving horizontally). The vertical component of the airspeed will be the full 250 km/h directed North.
The wind velocity is given as 65 km/h, blowing in the Southwest direction (225°). To determine its horizontal and vertical components, we can use trigonometry.
The horizontal component of the wind velocity will be 65 km/h multiplied by the cosine of the angle (225°).
Horizontal component of wind velocity = 65 km/h * cos(225°)
The vertical component of the wind velocity will be 65 km/h multiplied by the sine of the angle (225°).
Vertical component of wind velocity = 65 km/h * sin(225°)
Now, let's calculate the horizontal and vertical components of both the airspeed and wind velocity:
Airspeed:
Horizontal component of airspeed = 0 km/h
Vertical component of airspeed = 250 km/h
Wind velocity:
Horizontal component of wind velocity = 65 km/h * cos(225°)
Vertical component of wind velocity = 65 km/h * sin(225°)
Next, we can add the horizontal components of the airspeed and wind velocity together to find the horizontal component of the resultant velocity. Then, we can add the vertical components of the airspeed and wind velocity together to find the vertical component of the resultant velocity.
Horizontal component of resultant velocity = Horizontal component of airspeed + Horizontal component of wind velocity
Vertical component of resultant velocity = Vertical component of airspeed + Vertical component of wind velocity
Finally, we can use these components to calculate the magnitude and direction of the resultant velocity.
The magnitude of the resultant velocity can be found using the Pythagorean theorem:
Resultant velocity = sqrt((Horizontal component of resultant velocity)^2 + (Vertical component of resultant velocity)^2)
The direction of the resultant velocity can be calculated using the inverse tangent function:
Direction = atan(Vertical component of resultant velocity / Horizontal component of resultant velocity)
So, by calculating the above expressions, we can find the direction in which the pilot should head the plane, measured clockwise from North.