# Trajectory of a boat across a river- differential equations

• Apr 12th 2008, 04:33 PM
melystar
Trajectory of a boat across a river- differential equations
After a few hours coding and programming, I find that I have to create a differential equation. I don't know whether it's because my brains have already been pulverized but I just can't get my head around this!!

Boat enters a river of width, W. Tries to reach point P that is directly opposite from starting point S. Boat's skipper keeps boat pointing at P throughout journey. Angle between direction boat is pointing (P) and horizontal is theta. If boat's in still water, speed is V. River is flowing with speed v.

Equation at the point (x,y) I have this so far: dx/dt=V*cos(theta) and dy/dt=v-V*sin(theta) however I need to derive a function for dy/dx without any theta or trigonometric funcs and using only x, y, W, V and v. I have gotten up to here:

dy/dx=v/(V*cos(theta) - V*tan(theta)

I can't seem to find a way to relate W with anything else.
• Apr 13th 2008, 09:43 PM
CaptainBlack
Quote:

Originally Posted by melystar
After a few hours coding and programming, I find that I have to create a differential equation. I don't know whether it's because my brains have already been pulverized but I just can't get my head around this!!

Boat enters a river of width, W. Tries to reach point P that is directly opposite from starting point S. Boat's skipper keeps boat pointing at P throughout journey. Angle between direction boat is pointing (P) and horizontal is theta. If boat's in still water, speed is V. River is flowing with speed v.

Equation at the point (x,y) I have this so far: dx/dt=V*cos(theta) and dy/dt=v-V*sin(theta) however I need to derive a function for dy/dx without any theta or trigonometric funcs and using only x, y, W, V and v. I have gotten up to here:

dy/dx=v/(V*cos(theta) - V*tan(theta)

I can't seem to find a way to relate W with anything else.

What cordinate system are you using? Now draw a diagram, and a point (x,y).
Express tan(theta) and cos(theta) for a boat at this position in terma of x, y
and W aand then substitute into your differential equation.

RonL