# Problem: An airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 20° south of east. Its direction of motion relative to the Earth is 45.0° south of west, while its direction of travel relative to the air is 5.00° south of west.(a) What is the airplane’s speed relative to the air mass?(b) What is the airplane’s speed relative to the Earth?

###### FREE Expert Solution

Let the velocity of the airplane relative to the air be vpa, its velocity relative to the earth be vpg, and the velocity of the jet stream relative to the earth be vag.

The speed of the airplane relative to the earth is expressed as:

$\overline{){{\mathbf{v}}}_{\mathbf{p}\mathbf{g}}{\mathbf{=}}{{\mathbf{v}}}_{\mathbf{p}\mathbf{a}}{\mathbf{+}}{{\mathbf{v}}}_{\mathbf{a}\mathbf{g}}}$

Let east be the +x direction and north be the +y direction

The x components of velocity, vpg is given by:

vpgx = vpgcos(180°+45°) = vpg cos (225°)

Similarly, vpax = vpa cos(180°+5.00°) = vpa cos (185°)

and vagx = vagcos (-20°)

Therefore, the expression for the x components of the airplane's speed relative to the earth becomes:

$\overline{){{\mathbf{v}}}_{\mathbf{p}\mathbf{g}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\left(}}{\mathbf{225}}{\mathbf{°}}{\mathbf{\right)}}{\mathbf{=}}{{\mathbf{v}}}_{\mathbf{p}\mathbf{a}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\left(}}{\mathbf{185}}{\mathbf{°}}{\mathbf{\right)}}{\mathbf{+}}{{\mathbf{v}}}_{\mathbf{a}\mathbf{g}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\left(}}{\mathbf{-}}{\mathbf{20}}{\mathbf{°}}{\mathbf{\right)}}}$

Likewise, the y component of the airplane's speed relative to the earth is then written as:

$\overline{){{\mathbf{v}}}_{{\mathbf{pg}}}{\mathbf{sin}}{\mathbf{\left(}}{\mathbf{225}}{\mathbf{°}}{\mathbf{\right)}}{\mathbf{=}}{{\mathbf{v}}}_{{\mathbf{pa}}}{\mathbf{sin}}{\mathbf{\left(}}{\mathbf{185}}{\mathbf{°}}{\mathbf{\right)}}{\mathbf{+}}{{\mathbf{v}}}_{{\mathbf{ag}}}{\mathbf{sin}}{\mathbf{\left(}}{\mathbf{-}}{\mathbf{20}}{\mathbf{°}}{\mathbf{\right)}}}$

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###### Problem Details

An airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 20° south of east. Its direction of motion relative to the Earth is 45.0° south of west, while its direction of travel relative to the air is 5.00° south of west.
(a) What is the airplane’s speed relative to the air mass?
(b) What is the airplane’s speed relative to the Earth?