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

• Michael_0039
In summary, the student struggled with setting up the force balance equation and finding the acceleration, velocity, and position functions. They made a few mistakes, the most serious of which was removing the absolute value from equation (1), without first taking cases.
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.

Attachments

• Physics.pdf
2.6 MB · Views: 250
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 said:
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.

Attachments

• pic1.PNG
102.7 KB · Views: 312
But that is not showing your work on setting up the force balance equation and finding the acceleration, velocity, and position functions...

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.

Last edited:
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

Last edited:
Michael_0039

1. What is the formula for calculating the velocity of a body inside a liquid?

The formula for calculating the velocity of a body inside a liquid is v(x) = v0 * e-x/λ, where v0 is the initial velocity, x is the distance traveled, and λ is the characteristic length scale of the liquid.

2. How does the velocity of a body inside a liquid change as the distance traveled increases?

The velocity of a body inside a liquid decreases exponentially as the distance traveled increases. This is because the liquid exerts a drag force on the body, which increases with the velocity of the body. As the body travels further, the drag force becomes greater and slows down the body.

3. What is the significance of the characteristic length scale in the velocity equation?

The characteristic length scale, represented by λ in the velocity equation, is a measure of the distance over which the velocity decreases by a factor of e. It is an important parameter as it determines how quickly the velocity of the body decreases as it moves through the liquid.

4. What is the meaning of the term V(lim) in the velocity equation?

V(lim) represents the limiting velocity of the body inside the liquid. This is the maximum velocity the body can reach as it travels an infinite distance through the liquid. It is determined by the balance between the drag force and the force applied to the body.

5. How does the density of the liquid affect the velocity of a body inside it?

The density of the liquid does not directly affect the velocity of the body inside it. However, it does play a role in determining the drag force on the body, which in turn affects the velocity. A higher density liquid will generally result in a greater drag force and therefore a lower velocity for the body.

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