1-Dimensional Distance with Drag

[Matt Callicott]

Homework Statement

Find the distance, x, that an object of mass m travels through a medium, given an initial veloicty v₀.

Homework Equations

I searched the web and came across a site that showed how to deal with velocity dependent forces. http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node17.html#svelyd

Equations:
F_D = 1/2 ρCAv² , simplified to F_D = Bv²

mv (dv/dx) = -Bv²

The Attempt at a Solution

Going off of what the site said, I arrived at the following answer using these steps: mv dv = -Bv² dx
(1/B) v/v² dv = 1/m dx

-1/B ∫ 1/v dv = 1/m ∫ dx

-1/B ln(v / v₀) = x / m

So this gives me that x = -m/B ln(v / v₀)

However, this does not make since in the context of the question because x is dependent on v, which is not the initial velocity (v₀). I do not know if my interpretation is wrong or if I am doing something wrong here. Any help is appreciated. Thanks

 

[haruspex]

When you integrate, consider bounds.

 

[Matt Callicott]

After posting I did try to integrate with bounds. I tried v bounds of v₀ and v_f. Since I want to find the total distance, however, I substituted 0 as v_f. This gave me a problem with the ln function since ln(0) is undefined.

 

[haruspex]

After posting I did try to integrate with bounds. I tried v bounds of v₀ and v_f. Since I want to find the total distance, however, I substituted 0 as v_f. This gave me a problem with the ln function since ln(0) is undefined.

Ok, so how do you interpret that result?

 

[Matt Callicott]

Ok, so how do you interpret that result?

I’m not sure how to interpret that. Plugging in values here doesn’t yield a finite distance, yet I know that the object will eventually stop. What am I missing?

 

[haruspex]

yet I know that the object will eventually stop

Is that your intuition, or do you have some reasoning for that?

 

[Matt Callicott]

Is that your intuition, or do you have some reasoning for that?

My intuition tells me that it will stop; however, are you implying terminal velocity?

 

[haruspex]

are you implying terminal velocity?

No, there cannot be a nonzero terminal velocity since there is no force tending to maintain motion.
I am challenging your statement that it must stop after a finite distance.
On the one hand, your intuition tells you that, on the other, the equations say otherwise...

 

[Matt Callicott]

So the interpretation of the equation is that it takes an infinite distance to stop? I’m not following what you’re saying because my intuition is, what I have previously thought of as, strong that it will stop.

 

[Orodruin]

So the interpretation of the equation is that it takes an infinite distance to stop?

In your model where the drag is proportional to $v^2$, yes. That does not mean that this is a good physical model.

In general, for small velocities, the drag is proportional to $v$, not $v^2$.

 

[Matt Callicott]

In your model where the drag is proportional to v2v2v^2, yes. That does not mean that this is a good physical model.

In general, for small velocities, the drag is proportional to vvv, not v2v2v^2.

This is interesting. All of the equations that I have found are F_D=0.5pACv². Is there an alternative?

 

[Orodruin]

The $C$ in your equation is a constant only for large Reynold's numbers $R = vD/\nu$. For small Reynold's numbers, $C$ goes roughly as $1/R \propto 1/v$.

 

[rude man]

So the interpretation of the equation is that it takes an infinite distance to stop? I’m not following what you’re saying because my intuition is, what I have previously thought of as, strong that it will stop.

Beware of intuition! Physics is a mathematical science, not a subset of philosophy!

 

[haruspex]

Beware of intuition! Physics is a mathematical science, not a subset of philosophy!

Right, but equally beware assuming the same equation applies throughout the domain of interest. Sometimes intuition is telling you something important.

 

[Matt Callicott]

The $C$ in your equation is a constant only for large Reynold's numbers $R = vD/\nu$. For small Reynold's numbers, $C$ goes roughly as $1/R \propto 1/v$.

Sorry for the late response. Is there some mathematical function that describes the whole scope of the domain? A piece wise perhaps? What is the threshold in which you shift from proportionality of v² to v?

I’m having a difficult time comprehending that the force changes due to velocity. I’m not doubting you all’s expertise, I just don’t understand how you can do an analysis like this one with two equations and an ambiguous threshold.

Thank you all for your responses

 

[haruspex]

Sorry for the late response. Is there some mathematical function that describes the whole scope of the domain? A piece wise perhaps? What is the threshold in which you shift from proportionality of v² to v?

I’m having a difficult time comprehending that the force changes due to velocity. I’m not doubting you all’s expertise, I just don’t understand how you can do an analysis like this one with two equations and an ambiguous threshold.

Thank you all for your responses

There is no known equation of purely analytic form which operates at all speeds. If you were to create such an equation, analytic or otherwise, it would be found to be approximately quadratic at high speeds and linear at low speeds. There is a distinct transition over a fairly narrow range between linear (low) and turbulent (high) flow. Where exactly that transition occurs depends on parameters such as density and viscosity.

In the context of this thread, I note that you are given hardly any relevant values. This suggests to me that the question setter expects infinity as the answer. But as Orodruin points out, it must at some stage transition to linear, and that will result in a finite value, but you do not seem to have enough information to determine it.