Alright class, let's carefully analyze the problem:
We have a plane flying at a speed of [tex]\( 190 \, \text{m/s} \)[/tex] in the [tex]\( y \)[/tex]-direction. A wind is blowing at [tex]\( 25.0 \, \text{m/s} \)[/tex] in the [tex]\( x \)[/tex]-direction. We are to find the direction of the plane’s velocity with respect to the horizontal axis (the angle [tex]\( \theta \)[/tex]).
To determine this angle, we can use trigonometry, specifically the arctangent function, which relates the sides of a right triangle to the angles.
1. Identify the right triangle components:
- The adjacent side to the angle [tex]\( \theta \)[/tex] is the horizontal wind speed [tex]\( 25.0 \, \text{m/s} \)[/tex].
- The opposite side to the angle [tex]\( \theta \)[/tex] is the vertical plane speed [tex]\( 190 \, \text{m/s} \)[/tex].
2. Set up the arctangent function:
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. So, we have:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{25.0}{190}
\][/tex]
3. Calculate the angle [tex]\( \theta \)[/tex] in radians:
Using the arctangent function, [tex]\( \theta \)[/tex] is:
[tex]\[
\theta = \arctan\left(\frac{25.0}{190}\right)
\][/tex]
4. Convert the angle from radians to degrees:
Angles are often more conveniently expressed in degrees rather than radians. We use the conversion factor:
[tex]\[
1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees}
\][/tex]
After performing these steps, the angle [tex]\( \theta \)[/tex] is found to be approximately [tex]\( 0.13082739607405694 \)[/tex] radians. Converting radians to degrees, we get:
[tex]\[
\theta \approx 7.495857639729858 \, \text{degrees}
\][/tex]
So, the direction of the plane's velocity is approximately:
[tex]\[
\boxed{7.495857639729858} \, \text{degrees}
\][/tex]
Thus, the plane is flying at an angle of approximately [tex]\( 7.496 \)[/tex] degrees to the horizontal axis.
