To determine the magnitude of the plane's resultant velocity, we need to consider both the plane's velocity in the [tex]\( y \)[/tex]-direction and the wind's velocity in the [tex]\( x \)[/tex]-direction. These two velocities combine to form the resultant velocity, which can be visualized as the hypotenuse of a right-angled triangle where the other two sides represent the plane's and the wind's velocities.
We can use the Pythagorean theorem to find the magnitude of this resultant velocity. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ([tex]\( v \)[/tex]) is equal to the sum of the squares of the other two sides.
Given:
- The plane's velocity in the [tex]\( y \)[/tex]-direction ([tex]\( v_y \)[/tex]) = 190 m/s
- The wind's velocity in the [tex]\( x \)[/tex]-direction ([tex]\( v_x \)[/tex]) = 25.0 m/s
According to the Pythagorean theorem:
[tex]\[
v = \sqrt{v_y^2 + v_x^2}
\][/tex]
Now, substituting the given values into the equation:
[tex]\[
v = \sqrt{(190)^2 + (25.0)^2}
\][/tex]
[tex]\[
v = \sqrt{36100 + 625}
\][/tex]
[tex]\[
v = \sqrt{36725}
\][/tex]
Finally, solving for [tex]\( v \)[/tex]:
[tex]\[
v \approx 191.637678967368
\][/tex]
Therefore, the magnitude of the plane's resultant velocity is approximately [tex]\( 191.64 \, \text{m/s} \)[/tex].
