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I am studying a system where the velocity declines with time. The velocity change with time is well represented by hyperbolic decline equations of the form:

$$v = q_0\left(1+\frac{bt}{a}\right)^{-\dfrac 1b}$$

where v is the velocity and t is time. I want to solve for the parameter a. As I understand it, a is the initial decline. So I thought I could take the derivative of the hyperbolic decline equation, which is:

$$\frac{dv}{dt} = -\frac{q_0 \left(\frac{b t}{a}+1\right)^{-\frac{b+1}{b}}}{a}$$

I know the value of $q_0$; it is equal to the velocity at $t=0$. If I set $t = 0$, then the derivative is:

$$\frac{dv}{dt} = -\frac{q_0}{a}$$

But now, I'm stuck. Is the only way to continue on with determining a to actually use data? Is there anything else I can do?

At first I thought that a would equal $\frac{dv}{dt}$ and that $a^2=q_0$, but this doesn't result in a value that is even close, if actual data is used to determine a. By data I mean data in addition to $q_0$.

Actually, it looks like a needs to have units of "sec", so the idea that $a^2=q_0$, can't be right.

rdemyan
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1 Answers1

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Mathematically, $a$ is the ratio of the initial velocity to the initial deceleration. This is its only physical significance. You will need to use experimental data to determine the value of $a$, unless you have some additional information about the system other than the given velocity function.

To use experimental data to determine the value of $a$, you can plot a graph of velocity versus time. Then $a$ is simply the initial velocity divided by the negative of the initial slope.

Amogh
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