🤓 Based on our data, we think this question is relevant for Professor Rajeev's class at UR.

This problem requires us to determine the **direction** the plane should head so that it attains the desired direction.

Whenever given a problem about a plane and blowing wind, it's safe to assume it's a **two-dimensional relative motion** problem. The steps to solve a problem like this are going to be:

**Organize information (diagram):**the variables involved in a problem like this are velocities and directions.**Combine velocities.****Solve**for the target variable.

In general, the equation we'll use to add velocities is:

$\overline{){\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathit{P}\mathit{A}}{\mathbf{=}}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathit{P}\mathit{B}}{\mathbf{+}}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathit{B}\mathit{A}}}$

In this problem, the **plane** has some velocity compared to the **wind** and the **wind** has some velocity relative to the **ground**.

The plane needs to fly in a certain direction relative to the ground. The velocity will be the sum of the previous two vectors—let's call that ** v_{d}**, for "velocity in the desired direction," and

$\overline{){\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{{\mathit{d}}}{\mathbf{=}}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{{\mathit{p}}}{\mathbf{+}}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{{\mathit{w}}}}$

We're looking for the **angle** the plane has to head in order to fly 36° N of E. Let's call it ** θ_{p}**.

**Step 1:** Organize information (diagram)

Since the wind is blowing towards the south, the plane will have to adjust its heading towards the north to compensate. Therefore, ** θ_{p}** is greater than

If you find yourself stuck on this kind of problem or it’s starting to look terribly complicated, try changing the coordinate system to align with a different direction you know something about. To make the calculations easier for this problem, we'll rotate the coordinate system by ** θ_{d}** counterclockwise from the compass directions. After this rotation, the y-component of

Rotating the axes means that the wind blows at an angle *θ*_{d}__clockwise__ from the __negative y-axis__. Therefore:

$\overline{){{\mathit{v}}}_{\mathit{w}\mathit{x}}{\mathbf{=}}{\mathbf{-}}{{\mathit{v}}}_{{\mathit{w}}}{\mathbf{}}{\mathbf{sin}}{\mathbf{}}{{\mathit{\theta}}}_{{\mathit{d}}}\phantom{\rule{0ex}{0ex}}{{\mathit{v}}}_{{\mathit{w}}{\mathit{y}}}{\mathbf{=}}{\mathbf{-}}{{\mathit{v}}}_{{\mathit{w}}}{\mathbf{}}{\mathbf{cos}}{\mathbf{}}{{\mathit{\theta}}}_{{\mathit{d}}}}$

(** θ_{p}** −

$\overline{){{\mathit{v}}}_{{\mathit{p}}{\mathit{x}}}{\mathbf{=}}{{\mathit{v}}}_{{\mathit{p}}}{\mathbf{}}{\mathbf{cos}}{\mathbf{(}}{{\mathit{\theta}}}_{{\mathit{p}}}{\mathbf{-}}{{\mathit{\theta}}}_{{\mathit{d}}}{\mathbf{)}}\phantom{\rule{0ex}{0ex}}{{\mathit{v}}}_{{\mathit{p}}{\mathit{x}}}{\mathbf{=}}{{\mathit{v}}}_{{\mathit{p}}}{\mathbf{}}{\mathbf{sin}}{\mathbf{(}}{{\mathit{\theta}}}_{{\mathit{p}}}{\mathbf{-}}{{\mathit{\theta}}}_{{\mathit{d}}}{\mathbf{)}}}$

We have:

*v*_{wx} = −*v*_{w} sin(*θ*_{d}) = −82 sin (36°) = −48.2 km/h*v*_{wy} = −*v*_{w} cos(*θ*_{d}) = −82 cos (36°) = −66.3 km/h

*v*_{px} = ? = 560 cos(*θ*_{p} − *θ*_{d}) = 560 cos(*θ*_{p} − 36°)*v*_{py} = ? = 560 sin(*θ*_{p} − *θ*_{d}) = 560 sin(*θ*_{p} − 36°)

*v _{dx}* = ?

An airplane, whose air speed is 560 km/h, is supposed to fly in a straight path 36.0° N of E. But a steady 82 km/h wind is blowing from the north.

In what direction should the plane head?