Solving for Velocity: How to Integrate a Complex Function with Constants?

[Sewager]

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]

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!

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]

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?

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

 

[Ray Vickson]

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

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.

 

[Sewager]

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.

Thank you very much, I think I have a basic understanding now! Very appreciate it!