# What's the drag coefficient? [closed]

Suppose an objects starts with velocity $v$. A damping force that is dependent of velocity acts on this object ($F=-\beta v$). After traversing distance $l$, the object has velocity $\alpha v$ with $0<\alpha<1$. Calculate the drag coefficient $\beta$.

Well, it seems quite tricky. I tried starting with the dynamical equation: $$ma=m\frac{dv}{dt}=-\beta v$$which we can rearrange to get $$\frac{dv}{v}=-\frac{\beta}{m}dt$$We can integrate this: $$\int_{v}^{\alpha v}\frac{dv}{v}=-\int_{0}^{t}\frac{\beta}{m}dt$$However, the answer is dependent of time. I also tried writing $$ma=m\frac{dv}{dt}=m\frac{dv}{dx}\frac{dx}{dt}=m\frac{dv}{dx}v$$ and the dynamical equation becomes: $$m\frac{dv}{dx}=-\beta$$ we can rearrange this: $$mdv=-\beta dx$$ but again, after integrating, the answer is dependent in turn on the distance the object traversed.

Any hints?

## closed as off-topic by Qmechanic♦Jun 19 '17 at 19:59

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• The velocity will get any value in $(0,v]$. This problem can't be solved. – AHB Jun 19 '17 at 19:39
• OMG. I didn't see "After traversing distance l" ! And yes. It can be easily solved. Sorry for giving uneducated comment. – AHB Jun 19 '17 at 20:08
• The question was edited by the OP to replace "after an unknown time" to "after traversing a distance $l$. The new problem can be solved, the old one cannot. – mike stone Jun 20 '17 at 13:53

The solution to your equation is is $v(t)=v(0) \exp\{-\beta t/m\}$ as may be verified by plugging it in.
• Aaaand? It's dependent on time. The task is to express $\beta$ as a function of $v$ and $\alpha$ only. – Mermon Jun 19 '17 at 19:29
• @Mermon In the part you quoted of the question it doesn't specify that it needs to be a function of only $v$ and $\alpha$. Are you sure you can't have $t$ in the answer? This question is about a transient response, without using time we can really solve it. – JMac Jun 19 '17 at 20:00
• Well, and if we add that the body traversed distance $l$? – Mermon Jun 19 '17 at 20:02