2D Relative Motion with Distances and Times

When a boat crosses a river, the time to reach the opposite bank depends only on the perpendicular component of its velocity relative to the ground. To minimize crossing time, we maximize the velocity component perpendicular to the river banks.

Recall, using our relative velocity notation, we have that:

$\vec{v}_{b/g} = \vec{v}_{b/w} + \vec{v}_{w/g}$

where:

• $\vec{v}_{b/g}$ is the boat's velocity relative to ground
• $\vec{v}_{b/w}$ is the boat's velocity relative to water
• $\vec{v}_{w/g}$ is the water's velocity relative to ground

For a river of width $W$, the crossing time is:

$t = \dfrac{W}{v_{b/g,y}}$

where $v_{b/g,y}$ is the y-component (perpendicular to banks) of the boat's ground velocity.

The Shortest Time Strategy

For minimum crossing time, aim the boat perpendicular to the banks (at $\theta = 90°$). This maximizes the perpendicular velocity component:

...