- #1

doktorwho

- 181

- 6

## Homework Statement

On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose beginning is at ##O##. The boat 1 sails so that the vector of it's

*relative velocity*towards the water is always at ##\pi/2## with the line that connects the boat to the point ##O## and is constant ##v_f## in the direction of increasing angle.

At moment ##t=0## a boat 2 (M_0) is on the left side of the river at the location shown on the picture. It sails so that the vector of it's

*relative velocity*is always along the line that connects it to point ##O## and is constant ##v_m##. The velocity of river is ##v_0## and is also constant

If ##v_f=10v_0##, the width of river ##r_0##, determine:

a) trajectory of boat 1 ##r=f_f(θ)##

b) trajectory of boat 2 ##r=f_m(θ)##

c)what would be ##\frac{v_m}{v_0}## so that the two boats meet when the line that connects them makes an angle of ##θ=60°##

## Homework Equations

##\vec r = r*\vec e_r##

##\vec v =\dot r\vec e_r + r\dot θ\vec e_θ##

## The Attempt at a Solution

i)There are some things i don't get so i hope you can provide an insight into what is troubling me. I started with the boat 1 and tried to solve its trajectory:

##v_r=v_0cosθ## the radial component

##v_θ=v_f=v_0sinθ## the angle component

when i divide the equations i and integrate from ##\int_{r_0}^{r}## and ##\int_{θ_0}{θ}## i get ##r=r_0\frac{v_f/v_0 - sinθ_0}{v_f/v_0 - sinθ}##

ii)For the boat 2 same analysis is applied and i get the final trajectory to be:

##r=\frac{10r_0}{sinθ}[tan{\frac{θ}{2}}]^{v_m/v_0}## i was using the fact that we are integrating from the total width of the river to some ##r##, width is ##10r_0## and from the angle which was ##\pi/2## to some ##θ##

iii) The third part i don't seem to know how to start.. What exactly am i looking for here? What need to match? Their r's?