Modeling with First Order DFQ

Leaking Tank

Consider the setup below with a tank (volume $V$) that has some mass of $\text{ce}NaCl$, call it $m(t)$. A water with constant $\text{ce}NaCl$ concentration (0.25 kg/L) flows into the tank by a pipe at rate r L/min. To maintain constant volume water flows out at the same rate. Determine the $m(t)$ for a given initial condition $m_0$.

The hardest part is to write down the DFQ. Let’s to do it step by step. We have to ask ourselves how exactly $m(t)$ changes over time, or rather what is the instantaneous change $\mathrm{d}m/\mathrm{d}t$?

At any instant of time we know that an additional mass of $\text{ce}NaCl$ per time is $0.25\text{kg/L}\cdot r\text{L/min}=0.25r\text{kg/min}$. We also know that we lose $\text{ce}NaCl$ at a rate $m(t)/V\cdot r=m(t)r/V\text{kg/min}$. We combine what we know and get

$$\begin{array}{c}\frac{\mathrm{d}m}{\mathrm{d}t}=0.25r-m(t)r/V\\ \frac{\mathrm{d}m}{\mathrm{d}t}+m(t)r/V=0.25r\text{First Order DFQ}\end{array}$$

We already know the solution for this type of First Order Differential Equations (constant terms!).

$$m(t)=C{e}^{-(rt)/V}+0.25V$$

and then you solve the Initial Value Problem (IVP).

”True” Escape Velocity

A mass $m$ is projected upward with an initial velocity $v_0$. Find $v(x)$ and a maximum height $h$ it can reach. We know that the force on the mass is $F(x)=-GMm/(R+x{)}^2$ and using Newton’s laws:

$$\begin{array}{c}\cancel{m}a=-\frac{GM\cancel{m}}{(R+x{)}^2}\\ v\cdot \frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{GM}{(R+x{)}^2}\\ {\int }_{v_0}^{v}v\mathrm{d}v=-{\int }_0^x\frac{GM}{(R+x{)}^2}\mathrm{d}x\\ \frac{v^2}{2}-\frac{v_0^2}{2}=\frac{GM}{R+x}{|}_0^x\\ \frac{v^2}{2}-\frac{v_0^2}{2}=\frac{GM}{R+x}-\frac{GM}{R}\\ v^2=2GM(\frac{1}{R+x}-\frac{1}{R})+v_0^2\end{array}$$

Recall that normally, we would get $v^2=v_0^2-2gx$, which you can derive if you performTaylor expansion of the first term above. The graphs for each case would look like:

As you can see, with initial velocity of 11 km/s (Why?) graphs look the same in the beginning, but they diverge for larger distances! One can improve this model by adding damping term of $-\gamma \cdot v$, which would be a Second Order Differential Equation