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 ?, given the magnitude, |v[sub]WL[/sub]|, of the water's velocity relative to the land, and the maximum speed, |v[sub]BW[/sub]|, 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,
v[sub]BL[/sub] = v[sub]BW[/sub]+ v[sub]WL[/sub] .
If the boat is to travel straight across the river, i.e., along the y axis, then we need to have v[sub]BL[/sub],x=0.
This x component equals the sum of the x components of the other two vectors,
v[sub]BL,x[/sub] = v[sub]BW,x[/sub] + v[sub]WL,x[/sub] , or 0 = -|v[sub]BW[/sub]| sin ? + |v[sub]WL[/sub]| .
Solving for ?, we find sin?=|v[sub]WL[/sub]|/|v[sub]BW[/sub]|,
so ? =sin[sup]-1[/sup] |v[sub]WL[/sub]|/|v[sub]BW[/sub]|.
The following simulation let you play with it. Enjoy!
You can adjust the velocity of the river or the boat with slider.
You can also change it's direction (angle ?=c).
It will remember the last 3 traces.