Question

The nine ring wraiths want to fly from barad-dur to rivendell. rivendell is directly north of barad-dur. the dark tower reports that the wind is coming from the west at 69 miles per hour. in order to travel in a straight line, the ring wraiths decide to head northwest. at what speed should they fly (omit units)?

Answer 1

Velocity of the ring wraiths relative to the air:

$\vec v_{W/A}=(v_{W/A}\cos45^\circ,v_{W/A}\sin45^\circ)=\left(\dfrac{v_{W/A}}{\sqrt2},\dfrac{v_{W/A}}{\sqrt2}\right)$

The speed $v_{W/A}$ is what we want to find - this is the speed at which the wraiths are flying in the air.

(Note: $\vec v$ denotes velocity, while $v$ denotes speed)

Velocity of air relative to the ground:

$\vec v_{A/G}=((69\,\mathrm{mph})\cos180^\circ,(69\,\mathrm{mph})\sin180^\circ)=(-69,0)\,\mathrm{mph}$

The ring wraiths want their trajectory to be due north, which means their velocity relative to the ground should be

$\vec v_{W/G}=(v_{W/G}\cos90^\circ,v_{W/G}\sin90^\circ)=(0,v_{W/G})$

To an observer on the ground, $v_{W/G}$ is the speed at which the wraiths would appear to be moving in the air.

The relative velocities satisify

$\vec v_{W/A}+\vec v_{A/G}=\vec v_{W/G}$

$\implies\begin{cases}\dfrac{v_{W/A}}{\sqrt2}-(69\,\mathrm{mph})=0\\\\\dfrac{v_{W/A}}{\sqrt2}=v_{W/G}\end{cases}$

$\implies v_{W/A}=69\sqrt2\,\mathrm{mph}\approx98\,\mathrm{mph}$