Chapter 6 - Section 6.6 - Vectors - Exercise Set - Page 784: 113

The plane’s true speed relative to the ground is $269.08\ \text{miles/hr}$ and its compass heading relative to the ground is $278.34{}^\circ $.

Let the velocity of the plane with respect to earth be ${{\mathbf{v}}_{1}}$. The velocity of the plane with respect to wind is ${{\mathbf{v}}_{2}}$. The velocity of wind with respect to earth is $\mathbf{u}$. The formula to calculate the speed of the plane with respect to earth is ${{\mathbf{v}}_{1}}=\mathbf{u}+{{\mathbf{v}}_{2}}$ The speed of the plane with respect to the wind is $\begin{align} & {{\mathbf{v}}_{2}}=\left\| {{\mathbf{v}}_{2}} \right\|\cos \theta \mathbf{i}+\left\| {{\mathbf{v}}_{2}} \right\|\sin \theta \mathbf{j} \\ & =240\cos 170{}^\circ \mathbf{i}+240\sin 170{}^\circ \mathbf{j} \\ & =-236.35\mathbf{i}+41.67\mathbf{j} \end{align}$ Here, $\theta $ is the angle of the plane’s speed with respect to the wind from the horizontal axis in the anticlockwise direction. The speed of the wind with respect to the earth is $\begin{align} & \mathbf{u}=\left\| \mathbf{u} \right\|\cos \varphi \mathbf{i}+\left\| \mathbf{u} \right\|\sin \varphi \mathbf{j} \\ & =30\cos 185{}^\circ \mathbf{i}+30\sin 185{}^\circ \mathbf{j} \\ & =-29.88\mathbf{i}-2.61\mathbf{j} \end{align}$ Here, $\varphi $ is the angle of the wind’s speed with respect to earth from the horizontal axis in the anticlockwise direction. So, the speed of the plane with respect to the earth is $\begin{align} & {{\mathbf{v}}_{1}}=\mathbf{u}+{{\mathbf{v}}_{2}} \\ & =\left( -29.88\mathbf{i}-2.61\mathbf{j} \right)+\left( -236.35\mathbf{i}+41.67\mathbf{j} \right) \\ & =-266.23\mathbf{i}+39.06\mathbf{j} \end{align}$ The magnitude of the plane’s speed with respect to the earth is $\begin{align} & {{\mathbf{v}}_{1}}=\sqrt{{{\left( -266.23 \right)}^{2}}+{{\left( 39.06 \right)}^{2}}} \\ & =269.08\ \text{miles/hr} \end{align}$ The direction of a plane’s speed with respect to the earth is $\begin{align} & \tan \phi =\frac{39.06}{-266.23} \\ & =188.34{}^\circ \end{align}$ Here, $\phi $ is the angle of the plane’s speed with respect to the earth from the horizontal axis in the anticlockwise direction. Therefore, the compass heading relative to the ground is $270{}^\circ +8.34{}^\circ =278.34{}^\circ $.