Question

Construct Your Own Problem Consider an airplane headed for a runway in a cross wind. Construct a problem in which you calculate the angle the airplane must fly relative to the air mass in order to have a velocity parallel to the runway. Among the things to consider are the direction of the runway, the wind speed and direction (its velocity) and the speed of the plane relative to the air mass. Also calculate the speed of the airplane relative to the ground. Discuss any last minute maneuvers the pilot might have to perform in order for the plane to land with its wheels pointing straight down the runway.

Short answer

• $69.3\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$
  
• $81.8^{\circ }.$
  

Step-by-step solution

Definition of velocity

Velocity is the rate of change in position of an item in motion as seen from a specific frame of reference and measured by a specific time standard.

The velocity of the plane relative to the earth will be considered resultant.

This problem is the relative velocity problem; we can use the relative velocity formula.

Velocity of the plane relative to earth

Hence the velocity can be calculated by the following equation.

$V_{pe}=V_{pa}+V_{ae}$

The component table will be as below:

	X	Y
Vpa	$\left(-70\right)\cos \theta$	$\left(-70\right)\sin \theta$
Vae	10	0
Resultant	0 (plane is landing parallel to runway)	$\left(-70\right)\sin \theta$

Here we need to find the x component and the y component.

$$\begin{gathered} Xcomponent \\ {V}_{pax}=V_i\cos \theta \\ {V}_{pax}=\left(-70\right)\cos \theta \\ Ycomponent \\ {V}_{pay}=V_i\sin \theta \\ {V}_{pay}=\left(-70\right)\sin \theta \end{gathered}$$

Hence, solving the resultant equation to get the value of theta:

$\begin{array}{rcl}\left(-70\right)\cos \theta +10& =& 0\\ \left(70\right)\cos \theta & =& 10\\ \cos \theta & =& \frac{10}{70}\\ \theta & =& 81.8^{°}.\end{array}$

The angle of the direction will be$81.8^{°}$ .

Determine the resultant velocity

Hence the resultant vector will be:

$\begin{array}{rcl}\stackrel{\rightharpoonup }{R}& =& \sqrt{{\left(\sum _{}X\right)}^2+{\left(\sum _{}Y\right)}^2}\\ \stackrel{\rightharpoonup }{R}& =& \sqrt{{\left(0\right)}^2+{\left(-70\sin 81.8\right)}^2}\\ \stackrel{\rightharpoonup }{R}\left(V_{pe}\right)& =& 63.3\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\end{array}$

The velocity of the plane relative to the earth is $63.3\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.