Consider flying between Los Angeles and San Francisco, as shown in the image below. Assume that San Francisco is directly north of Los Angeles, and that the two cities are separated by a distance

D

$=$

$3$

$2$

$9$

m

i

l

e

s

D$=329$ miles. The engines of the plane $($as with any plane$)$ maintain the plane's speed relative to the air, which for the sake of this problem we take to be constant. Denoting the velocity of the plane relative to the air by

v

$$

p

$/$

a

$$v

$$

$$

$$p$/$a

$,$ this means that

$$

v

$$

p

$/$

a

$$

$=$

v

$0$

$=$

$5$

$0$

$7$

m

p

h

$$

$$v

$$

$$

$$p$/$a

$=$v

$0$

$=507$ mph where

v

$0$

$>$

$0$

v

$0$

$>0$ is constant. The air can also be moving with respect to the ground $($a phenomenon that we call wind$)$ with a velocity vector that we'll denote by

v

$$

a

$/$

g

$,$

$$v

$$

$$

$$a$/$g

$,$ thus making the plane's velocity vector relative to the ground,

v

$$

p

$/$

g

$$v

$$

$$

$$p$/$g

$,$ given by

v

$$

p

$/$

g

$=$

v

$$

p

$/$

a

$+$

v

$$

a

$/$

g

$$v

$$

$$

$$p$/$g

$=$

$$v

$$

$$

$$p$/$a

$+$

$$v

$$

$$

$$a$/$g

$$

Assuming there is no wind at all, how long does it take make a roundtrip from LA to SF and back?