Apostol question about the differential equations of a falling object

[zenterix]

Homework Statement

Refer to example 2 of section 8.6. Use the chain rule to write

$$\frac{dv}{dt}=\frac{ds}{dt}\frac{dv}{ds}=v\frac{dv}{ds}$$

and thus show that the differential equation in the example can be expressed as follows

$$\frac{ds}{dv}=\frac{bv}{c-v}$$

where $b=m/k$ and $c=gm/k$. Integrate this equation to express $s$ in terms of $v$. Check your result with the formulas for $v$ and $s$ derived in the example

Relevant Equations

The cited example 2 is a bit large to be shown here in all its steps.

Here are the main points and equations.

A body of mass $m$ is dropped from rest from a great height in the earth's atmosphere. Assume it falls in a straight line and that the only forces acting on it are the earth's gravitational attraction and a resisting force due to air resistance which is proportional to its velocity.

Newton's second law tells us

$$ma=mg-kv$$

where $k$ is some positive constant and $-kv$ is the force due to air resistance.

$$mv'=mg-kv$$

is a first-order equation in velocity $v$.

We can write this in the form

$$v'+\frac{k}{m}v=g$$

which we can solve using an integrating factor to obtain (assuming v(0)=0)

$$v(t)=e^{-kt/m}\int_0^t ge^{ku/m}du=\frac{mg}{k}(1-e^{-kt/m})$$

we can differentiate to find acceleration

$$a(t)=ge^{-kt/m}$$

We can also integrate to obtain position

$$s(t)=\frac{mg}{k}t+\frac{gm^2}{k^2}e^{-kt/m}+C$$

and if $s(0)=0$ we have

$$s(t)=\frac{mg}{k}t+\frac{gm^2}{k^2}(e^{-kt/m}-1)$$

Here is my solution to this problem. Unfortunately, I can't check it because it is not contained in the solution manual.

$$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=v\frac{dv}{ds}$$

$$\frac{ds}{dv}=\frac{v}{v'}=\frac{v}{ge^{-kt/m}}$$

$$=\frac{\frac{m}{k}v}{\frac{gm}{k}e^{-kt/m}}$$

$$=\frac{\frac{m}{k}v}{\frac{gm}{k}-\frac{gm}{k}+\frac{gm}{k}e^{-kt/m}}$$

$$=\frac{\frac{m}{k}v}{\frac{gm}{k}-\frac{gm}{k}(e^{-kt/m}-1)}$$

$$=\frac{bv}{c-v}$$

My main question is about the integration of this expression to obtain $s$ in terms of $v$.

$$\int_0^v \frac{ds}{dv}dv=\int_{s(0)}^{s(v)} ds = s(v)-s(0)=\int_0^v\frac{bv}{c-v}dv$$

$$=bc(\ln{(c)}-\ln{(c-v)})-bv$$

$$s(v)=s(0)+bc(\ln{(c)}-\ln{(c-v)})-bv$$

I don't recall seeing this relationship very often and so I am not sure if this is correct. The problem says to check this result with the equations derived in the cited Example 2. But that example derived equations for $v$ and $s$ relative to $t$. How would I go about using those equations to check my result of $s$ as a function of $v$?

 

[pasmith]

I think the intention is to derive the ODE as $$ma = mv\frac{dv}{ds} = mg - kv$$ so that $$\frac{ds}{dv} = \frac{mv}{mg - kv} = \frac{(m/k)v}{(mg/k) - v} = \frac{bv}{c - v}.$$ Note that the solution to the earlier exercise can be written in the form $$\begin{split} v(t) &= c(1 - e^{-b/t}) \\ s(t) - s_0 &= ct - bv(t).\end{split}$$ To show that $$s(v) - s_0 = bc\ln|c| - bv - bc\ln|c - v| = ct - bv = s(t) - s_0$$ the easiest way is to solve $v(t)$ for $t$ to obtain $$\begin{split} e^{-t/b} &= \frac{c - v}{c} \\ t &= b\ln|c| - b\ln|c - v|.\end{split}$$

 

[zenterix]

Attached are the full calculations leaving $v_0$ as a variable (just for added suffering with the algebra). Unfortunately, I couldn't figure out a way to post it directly here (file either too large or too low quality).