- #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?