A plane flies on a heading of N50 degrees East at a constant speed of 450 km/h. If the velocity of the wind is 70 km/h on a bearing of S30 degrees East,
what is the velocity (speed and direction) of the plane with respect to the ground?
a general solution to this problem...
a plane flies on a heading of θ with speed v with a wind blowing from the direction Ø with speed w.
The resultant vector z is
z = <zx,zy> = <v sinθ,v cosθ> - <w sinØ,w cosØ>
in the direction arctan(zx/zy)
Now plug in your numbers.
oops. arctan(zy/zx)
To find the velocity of the plane with respect to the ground, we need to add the velocities of the plane and the wind.
Step 1: Break down the velocities into their horizontal and vertical components.
The plane's velocity, 450 km/h on a heading of N50 degrees East, can be broken down into horizontal and vertical components. The horizontal component is given by 450 * cos(50) km/h, and the vertical component is given by 450 * sin(50) km/h.
The wind's velocity, 70 km/h on a bearing of S30 degrees East, can also be broken down into horizontal and vertical components. The horizontal component is given by 70 * cos(30) km/h, and the vertical component is given by 70 * sin(30) km/h.
Step 2: Add the horizontal and vertical components of the plane's and wind's velocities separately.
To find the horizontal component of the velocity of the plane with respect to the ground, add the horizontal components of the plane's and wind's velocities. Similarly, add the vertical components to find the vertical component of the velocity.
Horizontal component of the plane's velocity = 450 * cos(50) km/h
Horizontal component of the wind's velocity = 70 * cos(30) km/h
Vertical component of the plane's velocity = 450 * sin(50) km/h
Vertical component of the wind's velocity = 70 * sin(30) km/h
Step 3: Find the resultant horizontal and vertical components.
Add the horizontal components of the plane's and wind's velocities to get the resultant horizontal component. Similarly, add the vertical components to get the resultant vertical component.
Resultant horizontal component = Horizontal component of plane's velocity + Horizontal component of wind's velocity
Resultant vertical component = Vertical component of plane's velocity + Vertical component of wind's velocity
Step 4: Calculate the magnitude and direction of the resultant velocity.
Use the resultant horizontal and vertical components to calculate the magnitude and direction of the resultant velocity. The magnitude of the resultant velocity can be found using the Pythagorean theorem:
Magnitude of the resultant velocity = sqrt(Resultant horizontal component^2 + Resultant vertical component^2)
The direction of the resultant velocity can be found using trigonometry:
Direction of the resultant velocity = arctan(Resultant vertical component / Resultant horizontal component)
Plug in the values into the equations to find the magnitude and direction of the resultant velocity.