Things Certain and Uncertain: Classroom Capsule

In this section we offer some suggestions on how Euler’s opus moribundum can be used in a contemporary mathematics or physics classroom. The most natural connection is to differential equations, with an emphasis on mathematical modeling and solving unsolvable differential equations through approximation. Other than the notion of the integrating factor, the background material needed to carry out the exercises discussed below is addressed in first-year calculus, and anyone who knows derivatives up to integration by parts can take on this work. Finally, this episode can also provide an historical flavor to an introductory physics course in mechanics, especially if the consideration of units (from the 18th century to SI) is included. Therefore, E579 (or its English translation) can be used as a basis for a special assignment or project in such courses.

Below is a suggested sequence of self-contained exercises that solve a differential equation for the velocity of the balloon and calculate its maximum height.

Step 1. Consider the differential equation developed by Euler, with buoyancy ratio \(\lambda = 5\) and scale height \(k = 24,\!000\) pieds (presented here in modern notation):
\begin{eqnarray}
2v \frac{dv}{dx} + \left(\frac{1}{16} \cdot v^{2}e^{-x/k}\right) = 2g\left(5 e^{-x/k}-1\right),
\end{eqnarray}
where \(v = v(x)\) is the vertical velocity of the balloon as a function of its altitude \(x\), and \(g\) denotes the acceleration due to gravity.

Task: Show that this equation is a first-order, linear differential equation in the variable \(y = v^2\).

Step 2. Given the air pressure function \(f(x) = e^{-x/k}\) and its approximation \(g(x) = 1-\frac{x}{k}\), we can measure the relative error of the approximation via
\[
\text{RE} \;=\; \frac{g(x) - f(x)}{f(x)}.
\]

Tasks: Calculate the relative error at 1000 feet and at 10,000 feet. Based on your results, provide a range of altitudes for which you would consider \(g(x)\) to be a reasonable approximation for air pressure.

Step 3. Now consider the differential equation obtained after approximating the air pressure:
\[
2v \frac{dv}{dx} + \frac{1}{16}v^2 \;=\; 4g \left(4-\frac{5x}{k}\right).
\]

Task: Again using the variable \(y = v^2\), solve this differential equation using the integrating factor method.

Step 4. Once we know the solution \(y(x)\) from Step 3, we can easily find the balloon's maximum velocity.

Tasks: Explain why the maximum velocity will occur when \(y'(x) = 0\). Then, using \(k = 24,\!000\) pieds, find the maximum velocity attained by the balloon.