An airplane takes off at a ground velocity of 200 km/h toward the east, climbing at an angle of 14 degrees. A 20-km/h wind is blowing from the north. Determine the resultant air and ground velocities and their magnitudes.
r=w+p
w=[0,-20,0]
p=[200,0,200*tan14]
|r|=207
To determine the resultant air and ground velocities, we can break down the velocities into their horizontal and vertical components.
1. Determine the horizontal and vertical components of the airplane's velocity:
The horizontal component of the airplane's velocity is given by:
Horizontal component = Ground velocity * cos(angle)
Horizontal component = 200 km/h * cos(14 degrees)
The vertical component of the airplane's velocity is given by:
Vertical component = Ground velocity * sin(angle)
Vertical component = 200 km/h * sin(14 degrees)
2. Determine the horizontal and vertical components of the wind velocity:
Since the wind is blowing from the north, the horizontal component of the wind velocity is 0 km/h, and the vertical component is -20 km/h (negative because it is a southward wind).
3. Determine the resultant horizontal and vertical velocities:
The horizontal component of the resultant velocity is given by the sum of the airplane's horizontal velocity component and the wind's horizontal velocity component:
Resultant horizontal velocity = Airplane's horizontal velocity + Wind's horizontal velocity
Resultant horizontal velocity = Horizontal component of airplane's velocity + Horizontal component of wind velocity
The vertical component of the resultant velocity is given by the sum of the airplane's vertical velocity component and the wind's vertical velocity component:
Resultant vertical velocity = Airplane's vertical velocity + Wind's vertical velocity
Resultant vertical velocity = Vertical component of airplane's velocity + Vertical component of wind velocity
4. Determine the magnitude of the resultant air velocity:
Resultant air velocity = square root of (Resultant horizontal velocity^2 + Resultant vertical velocity^2)
5. Determine the magnitude of the resultant ground velocity:
The magnitude of the resultant ground velocity will be the same as the magnitude of the airplane's ground velocity since there is no horizontal wind component affecting the ground velocity.
Now, plug in the given values and calculate the answer.