2D Relative Motion: Resultant and Heading Angles

To continue illustrating the concept of 2D relative motion, let's now discuss a boat attempting to cross a flowing river.

When a boat crosses a river with a current, it experiences two simultaneous motions:

• The boat moves through the water at velocity $\vec{v}_{b/w}$ (b/w means boat with respect to water).
• The water carries the boat downstream at velocity $\vec{v}_{w/g}$ (w/g means water with respect to ground). This is the velocity of the river current.

The boat's actual path (relative to the ground) combines both motions:

Or, using better relative velocity notation:

Breaking into Components

To solve problems systematically, we break each velocity into x and y components:

Set up coordinates:

• x-direction: along the river (downstream positive)
• y-direction: across the river

River velocity: Only flows along x-axis

• $v_{w/g,x} = \text{river speed}$
• $v_{w/g,y} = 0$

Boat velocity: Depends on heading angle $\theta$

• $v_{b/w,x} = v_{boat} \cos(\theta)$
• $v_{b/w,y} = v_{boat} \sin(\theta)$

where $\theta$ is measured from the shore (the $+x$ axis).

Resultant velocity: Add corresponding components

• $v_{b/g,x} = v_{b/w,x} + v_{w/g,x}$
• $v_{b/g,y} = v_{b/w,y}$