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?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Relative Velocity in 2D concept. You can view video lessons to learn Relative Velocity in 2D. Or if you need more Relative Velocity in 2D practice, you can also practice Relative Velocity in 2D practice problems.

What professor is this problem relevant for?

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