Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

# Velocity vectors in relative motion

You wish to cross a river and arrive at a dock that is directly across from you, but the river's current will tend to carry you downstream. To compensate, you must steer the boat at an angle. Find the angle θ, given the magnitude, |vWL|, of the water's velocity relative to the land, and the maximum speed, |vBW|, of which the boat is capable relative to the water.

The boat's velocity relative to the land equals the vector sum of its velocity with respect to the water and the water's velocity with respect to the land,

vBL = vBW + vWL .

If the boat is to travel straight across the river, i.e., along the y axis, then we need to have vBL,x = 0. This x component equals the sum of the x components of the other two vectors,

vBL, x = vBW, x + vWL, x ,

or

0 = -|vBW| sin θ + |vWL| .

Solving for θ, we find

sin θ = |vWL|/|vBW| ,

so

Last Update: 2010-11-11