Vector Addition for Airplane Velocity Calculation
To solve this problem, you need to use vector addition to calculate the resultant velocity of the airplane relative to the ground, taking into account the wind's effect. The airplane has an initial velocity of 70 m/s, and the wind has a velocity of 18 m/s at a 120° angle to the airplane's direction.
First, let's define the vectors:
- The airplane's velocity vector, \( \vec{V}_a \), is 70 m/s in the direction of the positive x-axis.
- The wind's velocity vector, \( \vec{V}_w \), is 18 m/s at a 120° angle from the airplane's direction (which means it is 30° above the negative x-axis, or equivalently, 150° from the positive x-axis).
Now, decompose \( \vec{V}_w \) into x and y components:
- \( V_{wx} = \vec{V}_w \cos(150°) = 18 \cos(150°) = -18 \cos(30°) \) (since cos(150°) = -cos(30°))
- \( V_{wy} = \vec{V}_w \sin(150°) = 18 \sin(150°) = 18 \sin(30°) \) (since sin(150°) = sin(30°))
Calculate these components:
- \( V_{wx} = -18 \times \cos(30°) = -18 \times \sqrt{3}/2 \)
- \( V_{wy} = 18 \times \sin(30°) = 18 \times 1/2 \)
Now, add the wind's components to the airplane's velocity to find the resultant velocity vector, \( \vec{V}_r \):
- \( V_{rx} = V_{ax} + V_{wx} = 70 + (-18 \times \sqrt{3}/2) \)
- \( V_{ry} = V_{ay} + V_{wy} = 0 + 18 \times 1/2 \)
Calculate the magnitude of the resultant velocity:
- \( |\vec{V}_r| = \sqrt{V_{rx}^2 + V_{ry}^2} \)
Lastly, you'll want to calculate the distance traveled over 6 minutes. You can convert the time to seconds (6 minutes * 60 seconds/minute = 360 seconds) and then use \( d = |\vec{V}_r| \times t \) to find the distance \( d \).
Plug in the values and calculate the numerical results.
