3.5 Addition of velocities  (Page 2/12)

Adding velocities: a boat on a river

Refer to [link] , which shows a boat trying to go straight across the river. Let us calculate the magnitude and direction of the boat's velocity relative to an observer on the shore, ${\mathbf{\text{v}}}_{\text{tot}}$ . The velocity of the boat, ${\mathbf{\text{v}}}_{\text{boat}}$ , is 0.75 m/s in the $y$ -direction relative to the river and the velocity of the river, ${\mathbf{\text{v}}}_{\text{river}}$ , is 1.20 m/s to the right.

Strategy

We start by choosing a coordinate system with its $x$ -axis parallel to the velocity of the river, as shown in [link] . Because the boat is directed straight toward the other shore, its velocity relative to the water is parallel to the $y$ -axis and perpendicular to the velocity of the river. Thus, we can add the two velocities by using the equations $v_{\text{tot}}=\sqrt{v_x^2+v_y^2}$ and $\theta ={\text{tan}}^{-1}(v_y/v_x)$ directly.

Solution

The magnitude of the total velocity is

where

and

Thus,

yielding

The direction of the total velocity $\theta$ is given by:

This equation gives

Discussion

Both the magnitude $v$ and the direction $\theta$ of the total velocity are consistent with [link] . Note that because the velocity of the river is large compared with the velocity of the boat, it is swept rapidly downstream. This result is evidenced by the small angle (only $\mathrm{32.0º}$ ) the total velocity has relative to the riverbank.