Question

1. A duck is heading due north across a river with a speed of 12.0 km/h relative to the water. The water in the river has a uniform velocity of 6.00 km/h due east relative to the ground. Determine the velocity of the duck relative to an observer standing on either bank.

Answer 1

The total velocity of the duck relative to an observer standing on either bank is the vector sum of its velocity relative to the river plus the velocity of the river relative to the bank. So, we can find its magnitude from the Pythagorean theorem:

$$v_{tot}=\sqrt{v_{river}^2+v_{duck}^2},$$ $$v_{tot}=\sqrt{(6\ \dfrac{km}{h})^2+(12\ \dfrac{km}{h})^2}=13.42\ \dfrac{km}{h}.$$

We can find the direction of the total velocity of the duck from the geometry:

$$\theta=tan^{-1}(\dfrac{v_{duck}}{v_{river}}),$$ $$\theta=tan^{-1}(\dfrac{12\ \dfrac{km}{h}}{6\ \dfrac{km}{h}})=63.4^{\circ}.$$

Answer:

$v_{tot}=13.42\ \dfrac{km}{h}$, $63.4^{\circ} N\ of\ E$.