# Solving for the trajectory in the polar coordinate system

• doktorwho
In summary, the two boats have different trajectories when they meet. Boat 1 has a trajectory that is radial and makes an angle of 10 degrees with the x-axis. Boat 2 has a trajectory that is parallel and makes an angle of 60 degrees with the x-axis.
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?

doktorwho said:
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.

##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" ?

TSny said:
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.

BvU said:
##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?

doktorwho said:
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.

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

TSny said:
In the statement of the problem, you state that the width of the river is ro, but in your solution you take the width to be 10ro.
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?

Yes, your work looks correct to me.

doktorwho

## 1. What is the polar coordinate system and how is it used to solve for trajectory?

The polar coordinate system is a mathematical system used to represent points in a two-dimensional space. It uses a distance from the origin, known as the radius, and an angle from a fixed reference direction, known as the polar angle, to describe the location of a point. This system is often used in physics to represent the trajectory of an object, as it allows for the calculation of both distance and direction.

## 2. How do you convert Cartesian coordinates to polar coordinates in order to solve for trajectory?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), you can use the equations r = √(x² + y²) and θ = tan⁻¹(y/x). The radius r represents the distance from the origin to the point, and the angle θ represents the direction of the point from the origin.

## 3. What factors affect the trajectory of an object in the polar coordinate system?

The trajectory of an object in the polar coordinate system is affected by its initial speed, initial angle, and the force acting upon it. Other factors such as air resistance and gravity may also affect the trajectory.

## 4. How can the polar coordinate system be used to analyze projectile motion?

In projectile motion, an object is launched at an initial velocity and then follows a curved path due to the forces acting upon it. The polar coordinate system can be used to analyze this motion by breaking it down into its radial and tangential components. The radial component represents the distance from the origin, while the tangential component represents the change in angle over time.

## 5. Are there any limitations to using the polar coordinate system to solve for trajectory?

While the polar coordinate system is useful for solving for trajectory, it does have some limitations. It is not as intuitive to visualize as the Cartesian coordinate system, and it may be more difficult to work with in certain situations. Additionally, it may not be the best system to use for objects with complex trajectories, such as those affected by wind or air resistance.

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