Solving for the trajectory in the polar coordinate system

[doktorwho]

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?

 

[TSny]

1 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

Did you mean the second "=" in the second equation to be something else?

i get $r=r_0\frac{v_f/v_0 - sinθ_0}{v_f/v_0 - sinθ}$

OK

ii)For the boat 2 same analysis is applied and i get the final trajectory to be:
$r=\frac{10r_0}{sinθ}[tan{\frac{θ}{2}}]^{v_m/v_0}$

OK

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?

When the boats meet there will be one radial line from $O$ to both boats. I think you want that radial line to make a 60⁰ angle to the x-axis. But I might be misinterpreting the question.

 

[BvU]

$v_r=v_0\cos\theta$ gives boat F a velocity vector that makes an angle of $\pi\over 2$ with $\vec r$ as seen from the shore. But is that the "relative velocity towards the water" ?

 

[doktorwho]

Did you mean the second "=" in the second equation to be something else?
OK
OK
When the boats meet there will be one radial line from $O$ to both boats. I think you want that radial line to make a 60⁰ angle to the x-axis. But I might be misinterpreting the question.

$v_r=v_0\cos\theta$ gives boat F a velocity vector that makes an angle of $\pi\over 2$ with $\vec r$ as seen from the shore. But is that the "relative velocity towards the water" ?

Yeah, i meant $-$ instead of $=$
And as for the relative velocity, yes it is, in the diagram only the real vectors are drawn and for the forst boat the only radial component of that of $v_0$ hence even though the boat appears to go in circles it actually drifts a little radialy outward couse of that force so relaivly he has a radial component and that is its relative radial vector. Hos real one is 0.
As for the c) part, hmm.. i equalize the trajectory functions and make the angle 60?

 

[TSny]

As for the c) part, hmm.. i equalize the trajectory functions and make the angle 60?

Unfortunately, that would just be the point where the two trajectories cross. But, of course, that doesn't mean that both boats arrive at that point simultaneously. It's not clear that the two boats will ever meet at the same point. You would need to analyze the time dependence.

 

[TSny]

In the statement of the problem, you state that the width of the river is r_o, but in your solution you take the width to be 10r_o.

 

[doktorwho]

In the statement of the problem, you state that the width of the river is r_o, but in your solution you take the width to be 10r_o.

I apologize, i have mistakenly wrote $r_0$ instead of $10r_0$ and misgave the question, it asks for the the ratio when the TWO LINES meet, not necessaraly the boats.. sorry again but then the thing i said holds true?

 

[TSny]

Yes, your work looks correct to me.