Question 2.SP.33: A boat is traveling 20 km/h due east. An observer is station......
A boat is traveling 20 km/h due east. An observer is stationed 100 m south of the line of travel. Determine the angular velocity of the boat relative to the observer when in the position shown in Fig. 2-24.
Select unit vectors \mathbf{i} and \mathbf{j} in the east and north directions, respectively. Let \mathbf{r} be the position vector of the boat relative to the observer O. Then
\mathbf{r}=x \mathbf{i}+100 \mathbf{j}=100 \tan \theta \mathbf{i}+100 \mathbf{j}
The velocity \mathbf{v} of the boat is
\mathbf{v}=\dot{\mathbf{r}}=\frac{100 \dot{\theta}}{\cos ^{2} \theta} \mathbf{i}+0 \mathbf{j}
Since the speed v=40 \mathrm{~km} / \mathrm{h}=11.11 \mathrm{~m} / \mathrm{s} and \mathbf{\theta}=30^{\circ}, we have
11.11=\frac{100 \dot{\mathbf{\theta}}}{\cos ^{2} 30^{\circ}} \quad \text { or } \quad \mathbf{\omega}=\dot{\mathbf{\theta}}=\underline{0.0833 \mathrm{rad} / \mathrm{s}} \quad \text { clockwise}