Undergraduate physics: Body flows inside liquid [ v(x)=? and V(lim)=? ]

[Michael_0039]

Homework Statement

Object inside liquid

Relevant Equations

T=k*v^2
F=m*a

Homework Statement: Object inside liquid
Homework Equations: T=k*v^2
F=m*a

We hold an object with a mass (m) inside a liquid. On t=0 we free the object. Except the weight there is another one force, the friction of the liquid, witch is T=k*v^2 ( v=instant speed, and k=constant > 0). Also, we assume Lift force of liquid = 0. Which is the v(t) and limit speed of the object ? Attachment a pdf with my notes, and a schematic.

 

[berkeman]

Welcome to the PF.

You started to work on the problem, but didn't get very far. Can you write the sum of the forces on the object? And that will lead you to an equation for the acceleration, which leads you to the velocity and position as functions of time. Please show your work. Thank you.

 

[Michael_0039]

Welcome to the PF.

You started to work on the problem, but didn't get very far. Can you write the sum of the forces on the object? And that will lead you to an equation for the acceleration, which leads you to the velocity and position as functions of time. Please show your work. Thank you.

 

[berkeman]

But that is not showing your work on setting up the force balance equation and finding the acceleration, velocity, and position functions...

 

[Michael_0039]

Thanks for your answer :) !
I will look it again, maybe Ι made calculus mistake. Following, is my try but my asnwer in page 7 is not the one I expected.

 

[Delta2]

Though your final solution is not wrong because it verifies the ODE you have done some mistakes in the process.
One mistake is that you don't consider the constant of integration. And from page 5 and after you start doing serious algebraical mistakes. The most serious mistake is that you remove the absolute value, WITHOUT FIRST to take cases, regarding the velocity $u$.
In the following I have put $\alpha=\sqrt\frac{mg}{k}$ (and $c=(…)$, so continuing from 5 the correct step is
$$\frac{|u+\alpha|}{|u-\alpha|}=e^{ct}$$ (1)

Now consider cases
1) $\alpha>u>0$
2)$u=\alpha$
3)$u>\alpha$
and remove properly the absolute values in equation (1) in order to continue properly for each case

 

[Michael_0039]

Answer: https://docdro.id/tFieEuu

What's your view on this ?

Thanks.