Find a locus of points that define a relationship

[warfreak131]

Homework Statement

There is a collapsing gas cloud. You have a line of sight along z (the length of the triangle is x, the height is y), and you are measuring the line of sight velocity. There are pairs of points where the measured velocity is equal, even at different radii, due to the relationship between collapse velocity and radius.

The way my teacher described it was that at a greater radius from the center of the cloud, you have a larger x component, and therefore, will measure a certain velocity. But when you have a smaller radius, the speed of collapse is much greater, but the x component is much smaller, balancing the two scenarios out.

The question is

For the case that v(r) = v0 r^−1 , find equations for (r, θ), or
(x, y) where r = sqrt(x^2 + y^2) and θ = tan^−1 (x/y), of the locus of
points for which vr = 2v0 /Ri = constant. Here, Ri is the outer
radius of the collapsing region.

Homework Equations

The Attempt at a Solution

I equated the two velocities, v0 r^-1 = 2 v0/Ri, and I am left with r=Ri/2, but I don't know where that leaves me. I was also thinking of describing an ellipse with a minor axis of x/2, and a major axis of R, but I don't know how to incorporate the constant velocity, or my radius relationship.

 

[anathema]

Hello,

Thank you for sharing your question and attempting a solution. I am a scientist and I would be happy to help you with this problem.

Based on the information provided, it seems like you are on the right track with your approach. You have correctly equated the two velocities and solved for the radius, r = Ri/2. This means that at any point along this locus of points, the velocity vr will be constant at 2 v0/Ri. Now, we can use this information to find the equations for (r, θ) or (x, y).

Since we know that r = sqrt(x^2 + y^2), we can substitute r = Ri/2 into this equation to get:

Ri/2 = sqrt(x^2 + y^2)

Squaring both sides, we get:

Ri^2/4 = x^2 + y^2

This is the equation of a circle with radius Ri/2. We can also rewrite this equation in terms of θ by using the relation tan^-1 (x/y) = θ. This gives us:

tan^-1 (x/y) = tan^-1 (Ri/2y)

Solving for y, we get:

y = Ri/2 tan (θ)

This means that at any point along the locus of points, the value of y will be determined by the angle θ. Similarly, we can find the equation for x by using the relation r = sqrt(x^2 + y^2). This gives us:

x = sqrt(Ri^2/4 - y^2)

Substituting y = Ri/2 tan (θ), we get:

x = sqrt(Ri^2/4 - (Ri/2 tan (θ))^2)

Simplifying this equation, we get:

x = Ri/2 cos (θ)

This means that at any point along the locus of points, the value of x will also be determined by the angle θ.

In summary, the equations for (r, θ) or (x, y) for the locus of points where vr = 2v0/Ri = constant are:

r = Ri/2

θ = tan^-1 (Ri/2y)

x = Ri/2 cos (θ)

y = Ri/2 tan (θ)

I hope this helps to answer your question. Let me know if you have