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.
Implicit integration is a numerical method used to solve differential equations or systems of equations. It involves using an iterative process to find the solution, rather than directly solving the equations. It is commonly used in physics, engineering, and other scientific fields to model complex systems.
Explicit integration involves solving equations directly, while implicit integration uses an iterative process to find the solution. This means that implicit integration is more computationally intensive but can handle more complex equations that cannot be solved explicitly. It is also more stable for stiff equations, as it does not require a small time step to maintain accuracy.
Implicit integration can handle stiff equations, meaning equations with widely varying time scales, without sacrificing accuracy. It is also more robust and stable, and can handle a wider range of problems compared to explicit integration. Additionally, it can be used for higher-order differential equations, making it a versatile tool for complex systems.
Implicit integration is commonly used in physics, engineering, and other scientific fields to model and simulate complex systems. It is particularly useful for modeling physical systems with non-linear behavior, such as fluid dynamics, chemical reactions, and electrical circuits.
While implicit integration has many advantages, it also has some limitations. It is more computationally intensive and can be slower compared to explicit integration. It also requires an initial guess for the solution and may require more iterations to achieve convergence. Additionally, it may not be suitable for all types of equations, and the accuracy may depend on the initial guess and the chosen time step.