The discussion revolves around integrating a complex function for velocity, specifically the equation $$v = \sqrt{2g\frac{T-v \pi r^2t}{\pi R^2}}$$ where g, T, r, and R are constants. The goal is to find a relationship between velocity and time to derive a displacement equation.
Participants are actively engaging with the problem, offering suggestions on how to proceed with the integration. Some guidance has been provided on solving for velocity as a function of time and the potential complexity of the resulting quadratic equation.
There is an acknowledgment of the complexity involved in the integration process, and participants express a need for clarification on the setup and assumptions regarding the variables involved.
Sewager said:Homework Statement
Integrate
$$v = \sqrt{2g\frac{T-v \pi r^2t}{\pi R^2}}$$
where g,T,r,R are constants
Homework Equations
N/A
The Attempt at a Solution
I tried playing around with the variables, but I am not sure how to integrate this. Just give me a little bit of hint would do. Thanks!
Ray Vickson said:Integrate what? Are you solving for ##v##as a function of ##r## or ##t## for example, then calculating ##\int v\, dr## or ##\int v\, dt##? Or, are you solving for (say) ##r## as a function of ##v## then computing ##\int r \, dv##? Or are you trying to do something else?
Sewager said:I sincerely apologize for my lack of explanations.
v is velocity and t is time. The rest are just constants. I want to integrate velocity vs. time to find the displacement equation
Thank you very much, I think I have a basic understanding now! Very appreciate it!Ray Vickson said:It will be messy. Just solve for ##v## as a function of ##t## (I.e., ##v = f(t)##) then integrate, or try to. You will get a quadratic equation in ##v##, so there will be two roots (that is, two functions ##v = f_1(t)## or ##v = f_2(t)##) and you will need to figure out which one is the correct root, probably using other information that you have.