Using Standard Basis Vectors to Model a Problem

An airplane has an air speed of \(400\) km\(/\)h and is headed east. There is a northwesterly wind of \(80\) km\(/\)h. (Northwesterly winds blow toward the southeast.)

  1. Find a vector representing the velocity of the airplane in the air.
  2. Find a vector representing the wind velocity.
  3. Find a vector representing the velocity of the airplane relative to the ground.
  4. Find the true speed of the airplane.

Solution We use a two-dimensional coordinate system with the direction north along the positive \(y\)-axis. Then the direction east is along the positive \(x\)-axis.

Using a scale of \(1~\hbox{unit} = 1\) km\(/\)h, define \[ \begin{eqnarray*} \mathbf{v}_{\mathrm{a}} &=& \hbox{Velocity of the airplane in the air}\\[2pt] \mathbf{v}_{\mathrm{w}} &=& \hbox{Velocity of the wind}\\[2pt] \mathbf{v}_{\mathrm{g}} &=& \hbox{Velocity of the airplane relative to the ground} \end{eqnarray*} \]

Figure 31 shows the vectors \(\mathbf{v}_{a},\) \(\mathbf{v}_{w},\) \(\mathbf{v}_{g}.\)

  1. The vector \(\mathbf{v}_{\rm{a}}\) has magnitude \(400\) and direction \(\mathbf{i}\), so \(\mathbf{v}_{\mathrm{a}}\) = 400i.
  2. The vector \(\mathbf{v}_{\mathrm{w}}\) has magnitude \(80\) and makes an angle of \(\dfrac{7\pi}{4}\) with the positive \(x\)-axis, as shown in Figure 32. Since the magnitude of \(\mathbf{v}_{\rm{w}}\) is 80, \[ \mathbf{v}_{\rm{w}} = 80\left(\cos \dfrac{7\pi}{4}\mathbf{i} + \sin \dfrac{7\pi}{4}\mathbf{j}\right) = 80 \left(\dfrac{\sqrt{2}}{2}\mathbf{i} - \dfrac{\sqrt{2}}{2}\mathbf{j}\right) = 40\sqrt{2}\mathbf{i} - 40\sqrt{2}\mathbf{j} \]
  3. The velocity \(\mathbf{v}_{\mathrm{g}}\) of the airplane relative to the ground is the resultant of \(\mathbf{v}_{\rm{a }}\) and \( \mathbf{v}_{\rm{w}}.\) \[ \mathbf{v}_{\mathrm{g}} = \mathbf{v}_{\mathrm{a}} + \mathbf{v}_{\mathrm{w}} = 400\ \mathbf{i} + \left[ 40\sqrt{2} \mathbf{i} - 40\sqrt{2} \mathbf{j}\right] = (400 + 40\sqrt{2}) \mathbf{i} - 40\sqrt{2} \mathbf{j} \]
  4. The true speed of the airplane is the magnitude of the vector \(\mathbf{v}_{\mathrm{g}}.\) \[ \left\Vert \mathbf{v}_{\mathrm{g}}\right\Vert =\sqrt{(400 + 40\sqrt{2})^{2} + (40\sqrt{2})^{2}}\approx 460.060 \]

The true speed of the airplane is approximately \(460.060 \,\rm{km/h}\).