Prequel to “A Relentless Pursuit”

July 15, 2021August 18, 2021 / Michael Xue / Leave a comment

Fig. 1

Illustrated in Fig. 2 are two circular hoops of unit radius, centered on a common x-axis and a distance $2a$ apart. There is also a soap films extends between the two hoops, taking the form of a surface of revolution about the x-axis. If gravity is negligible, the film takes up a state of stable, equilibrium in which its surface area is a minimum.

Fig. 2

Our problem is to find the function $y(x)$ , satisfying the boundary conditions

$y(-a) = y(a) = 1,\quad\quad\quad(1)$

which makes the surface area

$A=2\pi\displaystyle\int\limits_{-a}^{a}y\sqrt{1+(y')^2}\;dx\quad\quad\quad(2)$

a minimum.

Let

$F(x,y, y') = 2\pi y \sqrt{1+(y')^2}.$

We have

$\frac{\partial F}{\partial y} = 2\pi \sqrt{1+(y')^2}$

and

$\frac{\partial F}{\partial y'} = 2 \pi y \cdot\frac{1}{2}\left(1+(y')^2\right)^{-\frac{1}{2}}\cdot 2y'=\frac{2 \pi y y'}{\sqrt{1+(y')^2}}.$

The Euler-Lagrange equation

$\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0$

becomes

$\sqrt{1+(y')^2} - \frac{d}{dx}\left(\frac{y y'}{\sqrt{1+(y')^2}}\right) = 0.$

Fig. 3

Using Omega CAS Explorer (see Fig. 3), it can be simplified to:

$y \frac{d^2 y}{dx^2}- \left(\frac{dy}{dx}\right)^2=1.$

This is the differential equation solved in “A Relentless Pursuit” whose solution is

$y = C_1\cdot \cosh(\frac{x+C_2}{C_1}).$

We must then find $C_1$ and $C_2$ subject to the boundary condition (1), i.e.,

$C_1\cdot \cosh(\frac{a+C_2}{C_1}) = C_1\cdot\cosh(\frac{-a+C_2}{C_1})\implies \cosh(\frac{a+C_2}{C_1}) = \cosh(\frac{-a+C_2}{C_1}).$

The fact that $\cosh$ is an even function implies either

$a+C_2 = -a+C_2\quad\quad\quad(3)$

$a+C_2 = -(-a+C_2).\quad\quad\quad(4)$

While (3) is clearly false since it claims for all $a >0, a = -a$ , (4) gives

$C_2=0.$

And so, the solution to boundary-value problem

$\begin{cases} y \frac{d^2 y}{dx^2}- \left(\frac{dy}{dx}\right)^2=1,\\ y(-a)=y(a)=1\end{cases}\quad\quad\quad(5)$

$y = C_1\cdot \cosh(\frac{x}{C_1}).\quad\quad\quad(6)$

To determine $C_1$ , we deduce the following equation from the boundary conditions that $y=1$ at $x=\pm a:$

$C_1\cdot \cosh(\frac{a}{C_1}) = 1.\quad\quad\quad(7)$

This is a transcendental equation for $C_1$ that can not be solved explicitly. Nonetheless, we can examine it qualitatively.

Let

$\mu = \frac{a}{C_1}$

and express (7) as

$\cosh(\mu) = \frac{\mu}{a}.\quad\quad\quad(8)$

Fig. 4

A plot of (8)’s two sides in Fig. 4 shows that for sufficient small $a$ , the curves $z = \cosh(\mu)$ and $z = \frac{\mu}{a}$ will intersect. However, as $a$ increases, $z=\frac{\mu}{a}$ , a line whose slope is $\frac{1}{a}$ rotates clockwise towards $\mu$ -axis. The curves will not intersect if $a$ is too large. The critical case is when $a=a^*$ , the curves touch at a single point, so that

$\cosh(\mu) = \frac{\mu}{a^*}\quad\quad\quad(9)$

and $y=\frac{\mu}{a}$ is the tangent line of $z=\cosh(\mu),$ i.e.,

$\sinh(\mu) = \frac{1}{a^*}.\quad\quad\quad(10)$

Dividing (9) by (10) yields

$\coth(\mu) = \mu. \quad\quad\quad(11)$

What the mathematical model (5) predicts then is, as we gradually move the rings apart, the soap film breaks when the distance between the two rings reaches $2a^*$ , and for $a > a^*$ , there is no more soap film surface connects the two rings. This is confirmed by an experiment (see Fig. 1).

We compute the value of $a^*$ , the maximum value of $a$ that supports a minimum area soap film surface as follows.

Fig. 5

Solving (11) for $\mu$ numerically (see Fig. 5), we obtain

$\mu = 1.1997.$

By (10), the corresponding value of

$a^* = \frac{1}{\sinh(\mu)} = \frac{1}{\sinh(1.1997)} = 0.6627$ .

We also compute the surface area of the soap film from (2) and (6) (see Fig. 6). Namely,

$A = 2\pi \displaystyle\int\limits_{-a}^{a} C_1 \cosh\left(\frac{x}{C_1}\right) \sqrt{1+\left(\frac{d}{dx}C_1\cosh\left(\frac{x}{C_1}\right)\right)^2}\;dx = \pi C_1^2\left(\sinh\left(\frac{2a}{C_1}\right) + \frac{2a}{C_1}\right).$

Fig. 6

Exercise-1 Given $a=\frac{1}{2}$ , solve (7) numerically for $C_1.$

Exercise-2 Without using a CAS, find the surface area of the soap film from (2) and (6).

That first sip of coffee in the morning

December 31, 2020January 2, 2021 / Michael Xue / Leave a comment

It is a good idea to enjoy a cup of coffee before starting a busy day.

Suppose the coffee fresh out of the pot with temperature $\alpha^{\circ} C$ is too hot, we can immediately add cream to reduce the temperature by $\delta^{\circ} C$ instantly, then wait for the coffee to cool down naturally to $\omega^{\circ} C$ before sipping it comfortably. We can also wait until the temperature of the coffee drops to $(\omega+\delta)^{\circ} C$ first, then add the cream to further reduce it instantly to $\omega^{\circ} C$ .

Typically, $\alpha = 90, \omega = 75$ , and $\delta = 5$ .

If we are in a hurry and want to wait the shortest possible time, should the cream be added right after the coffee is made, or should we wait for a while before adding the cream?

The heat flow from the hot water to the surrounding air obeys Newton’s cooling and heating law, described by the following ordinary differential equation:

$\frac{d}{dt}\theta(t) = k (E-\theta(t))$

where $\theta(t)$ , a function of time $t$ , is the temperature of the water, $E$ is the temperature of its surroundings, and $k>0$ is a constant depends on the heat transfer mechanism, the contact are with the surroundings, and the thermal properties of the water.

Fig. 1 a place where Newton’s law breaks down

Under normal circumstances, we have

$E \ll \omega < \omega+\delta < \alpha-\delta < \alpha\quad\quad\quad(1)$

Based on Newton’s law, the mathematical model of coffee cooling is:

$\begin{cases} \frac{d}{dt}\theta(t) = k (E-\theta(t)) \\ \theta(0)= \theta_0\end{cases}\quad\quad\quad(2)$

Fig. 2

Solving (2), an initial-value problem (see Fig. 2) gives

$\theta(t) = E + e^{-kt}(\theta_0 - E).$

Therefore,

$t = \frac{\log\left(\frac{E-\theta_0}{E-\theta(t)}\right)}{k}=\frac{\log\left(\frac{\theta_0-E}{\theta(t)-E}\right)}{k}.\quad\quad\quad(3)$

If cream is added immediately (see Fig. 3),

Fig. 3 : cream first

by (3),

$t_1=\frac{\log\left(\frac{(\alpha-\delta)-E}{\omega-E}\right)}{k}.$

Otherwise (see Fig. 4),

Fig. 4: cream last

$t_2=\frac{\log\left(\frac{\alpha-E}{(\omega+\delta)-E}\right)}{k}.$

And so,

$t_1-t_2 = \frac{\log\left(\frac{(\alpha-\delta)-E}{\omega-E}\right)}{k}- \frac{\log\left(\frac{\alpha-E}{(\omega+\delta)-E}\right)}{k}=\frac{1}{k}\left(\log\left(\frac{(\alpha-\delta)-E}{\omega-E}\right)-\log\left(\frac{\alpha-E}{(\omega+\delta)-E}\right)\right)\quad(4)$

Fig. 5

Since

$\frac{(\alpha-\delta)-E}{\omega-E}- \frac{\alpha-E}{(\omega+\delta)-E}=\frac{(\alpha-\delta-E)(\omega+\delta-E)-(\omega-E)(\alpha-E)}{(\omega-E)(\omega+\delta-E)}=\frac{-\delta(\omega+\delta-\alpha)}{(\omega-E)(\omega+\delta-E)}\overset{(1)}{>0}$

implies

$\log\left(\frac{(\alpha-\delta)-E}{\omega-E}\right)-\log\left(\frac{\alpha-E}{(\omega+\delta)-E}\right)>0,\quad\quad\quad(5)$

from (4) , we see that

$t_1-t_2 > 0;$

i.e.,

$t_1 > t_2$

Hence,

If we are in a hurry and want to wait the shortest possible time, we should wait for a while before adding the cream!

Exercise-1 Solve (2) without using a CAS.

Exercise-2 Show that $\frac{\alpha-\delta-E}{\omega-E}\cdot\frac{\omega+\delta-E}{\alpha-E} >1.$

An ODE to Thanksgiving

December 8, 2020August 2, 2021 / Michael Xue / Leave a comment

A turkey is taken from the refrigerator at ${\theta_0}^{\circ} C$ and placed in an oven preheated to $E^{\circ} C$ and kept at that temperature; after $t_1$ minutes the internal temperature of the turkey has risen to ${\theta_1}^{\circ} C$ . The fowl is ready to be taken out when its internal temperature reaches ${\theta_2}^{\circ} C$ .

Typically, ${\theta_0} = 2, E=200, t_1=30, \theta_1=16, \theta_2=88$ .

Determine the cooking time required.

According to Newton’s law of heating and cooling (see “Convective heat transfer“), the rate of heat gain or loss of an object is directly proportional to the difference in the temperatures between the object and its surroundings. This law is best described by the following ODE (Ordinary Differential Equation):

$\frac{d}{dt}{\theta(t)} = k\cdot(E-\theta(t)),\quad\quad\quad(1)$

where $\theta(t), E$ are the temperatures of the object and its surroundings respectively. $k > 0$ is the constant of proportionality.

Fig. 1

We see that (1) has a critical point $\theta^* = E$ . Fig. 1 illustrates the fact that depending on its initial temperature, an object either heats up or cools down, trending towards $E$ in both cases.

We formulate the problem as a system of differential-algebraic equations:

$\begin{cases} \frac{d}{dt}{\theta(t)} = k\cdot(E-\theta(t)) \\ \theta(0)=\theta_0\\ \theta(t_1)=\theta_1 \\ \theta(\boxed{t_2}) = \theta_2\end{cases}(2)$

To find the required cooking time, we solve (2) for $t_2$ (see Fig. 2).

Fig. 2

Using Omega CAS Explorer, the typical cooking time is found to be approximately $4$ hours ( $3.88...\approx 4$ )

Luise Lange of Woodrow Wilson Junior College once wrote (see ” A Century of Calculus, Part I”, p. 50):

“In many calculus texts problems are formulated too one-sidedly in terms of particular, numerical data rather than in general terms. While pedagogically it may be wise to begin a new type of problem with some numerical examples, it is only the general formulation, and the interpretation of the answer in general terms, which can give insight into the functional relation between the given and the derived data.”

I agree with her wholeheartedly! On encountering a mathematical modeling problem stated with numerical values, I prefer to re-state it using symbols first. Then solve the problem symbolically. The numerical values are substituted for the symbols at the very end.

This post is a case in point, as the problem is re-formulated from page 1005 of Jan Gullberg’s “Mathematics From the Birth of Numbers”:

Exercise-1 Solving (2) without using a CAS.

Exercise-2 Given $\theta_0< \theta_1 < E$ , show that

$k = \frac{t\log(\frac{E-\theta_0}{E-\theta_1})}{t_1} > 0.$

Exercise-3 Given $\theta_0 < \theta_1< E$ , verify that $\lim\limits_{t\rightarrow \infty} \theta(t) = E$ from

$\theta(t) = -Ee^{ -\frac{t\log(\frac{E-\theta_0}{E-\theta_1})}{t_1} } + \theta_0 e^{-\frac{t\log(\frac{E-\theta_0}{E-\theta_1})}{t_1}} + E.$

Exercise-4 A slice is cut from a loaf of bread fresh from the oven at $180^{\circ} C$ and placed in a room with a constant temperature of $20^{\circ} C$ . After 1 minute, the temperature of the slice is $140^{\circ} C$ . When has the slice of bread cooled to $32^{\circ} C$ ?

What is the shape of a hanging rope?

August 14, 2020August 31, 2020 / Michael Xue / Leave a comment

Question: What is the shape of a flexible rope hanging from nails at each end and sagging under the gravity?

Answer:

First, observe that no matter how the rope hangs, it will have a lowest point $A$ (see Fig. 1)

Fig. 1

It follows that the hanging rope can be placed in a coordinate system whose origin coincides with the lowest point $A$ and the tangent to the rope at $A$ is horizontal:

Fig. 2

At $A$ , the rope to its left exerts a horizontal force. This force (or tension), denoted by $T_0$ , is a constant:

Fig. 3

Shown in Fig. 3 also is an arbitrary point $B$ with coordinates $(x, y)$ on the rope. The tension at $B$ , denoted by $T_1$ , is along the tangent to the rope curve. $\theta$ is the angle $T_1$ makes with the horizontal.

Since the section of the rope from $A$ to $B$ is stationary, the net force acting on it must be zero. Namely, the sum of the horizontal force, and the sum of the vertical force, must each be zero:

$\begin{cases}T_1cos(\theta)=T_0\quad\quad\quad(1)\\ T_1\sin(\theta) = \rho gs\;\;\quad\quad(2)\end{cases}$

where $\rho$ is the hanging rope’s mass density and $s$ its length from $A$ to $B$ .

Dividing (2) by (1), we have

$\frac{T\sin(\theta)}{T\cos(\theta)} = \tan(\theta) = \frac{\rho g}{T_0}s\overset{k=\frac{\rho g}{T_0}}{\implies} \tan(\theta) = ks.\quad\quad\quad(3)$

Since

$\tan(\theta) = \frac{dy}{dx}$ , the slope of the curve at $B$ ,

and

$s = \int\limits_{0}^{x}\sqrt{1+(\frac{dy}{dx})^2}$ ,

we rewrite (3) as

$\frac{dy}{dx} = k \int\limits_{0}^{x}\sqrt{1+(\frac{dy}{dx})^2}\;dx$

and so,

$\frac{d^2y}{dx^2}=k\cdot \frac{d}{dx}(\int\limits_{0}^{x}\sqrt{1+(\frac{dy}{dx})^2}\;dx)=k\sqrt{1+(\frac{dy}{dx})^2}$

i.e.,

$\frac{d^2y}{dx^2}=k\sqrt{1+(\frac{dy}{dx})^2}.\quad\quad\quad(4)$

To solve (4), let

$p = \frac{dy}{dx}$ .

We have

$\frac{dp}{dx} = k\sqrt{1+p^2} \implies \frac{1}{\sqrt{1+p^2}}\frac{dp}{dx}=k.\quad\quad\quad(5)$

Integrate (5) with respect to $x$ gives

$\log(p+\sqrt{1+p^2}) = kx + C_1\overset{p(0)=y'(0)=0}{\implies} C_1 = 0.$

i.e.,

$\log(p+\sqrt{1+p^2}) = kx.\quad\quad\quad(6)$

Solving (6) for $p$ yields

$p = \frac{dy}{dx} =\sinh(kx).\quad\quad\quad(7)$

Integrate (7) with respect to $x$ ,

$y = \frac{1}{k} \cosh(kx) + C_2\overset{y(0)=0,\cosh(0)=1}{\implies}C_2=-\frac{1}{k}$ .

Hence,

$y = \frac{1}{k}\cosh(kx)-\frac{1}{k}$ .

Essentially, it is the hyperbolic cosine function that describes the shape of a hanging rope.

Exercise-1 Show that $\int \frac{1}{\sqrt{1+p^2}} dp = \log(p + \sqrt{1+p^2})$ .

Exercise-2 Solve $\log(p+\sqrt{1+p^2}) = kx$ for $p$ .

From Dancing Planet to Kepler’s Laws

May 12, 2020July 1, 2020 / Michael Xue / 2 Comments

“This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent powerful Being” – Sir. Issac Newton

When I was seven years old, I had the notion that all planets dance around the sun along a wavy orbit (see Fig. 1).

Fig. 1

Many years later, I took on a challenge to show mathematically the orbit of my ‘dancing planet’ . This post is a long overdue report of my journey.

Shown in Fig. 2 is the sun and a planet in a x-y-z coordinate system. The sun is at the origin. The moving planet’s position is being described by $x=x(t), y=y(t), z=z(t)$ .

Fig. 2 $r=\sqrt{x^2+y^2+z^2}, F=G\frac{Mm}{r^2}, F_z=-F\cos(c)=-F\cdot\frac{z}{r}$

According to Newton’s theory, the gravitational force sun exerts on the planet is

$F=-G\cdot M \cdot m \cdot \frac{1}{r^2}(\frac{x}{r},\frac{y}{r}, \frac{z}{r})=-\mu\cdot m \cdot \frac{1}{r^3}\cdot(x, y, z)$

where $G$ is the gravitational constant, $M, m$ the mass of the sun and planet respectively. $\mu = G\cdot M$ .

By Newton’s second law of motion,

$\frac{d^2x}{dt^2} = -\mu\frac{x}{r^3},\quad\quad\quad(0-1)$

$\frac{d^2y}{dt^2} = -\mu\frac{y}{r^3},\quad\quad\quad(0-2)$

$\frac{d^2z}{dt^2} = -\mu\frac{z}{r^3}.\quad\quad\quad(0-3)$

$y \cdot$ (0-3) $- z \cdot$ (0-2) yields

$y\frac{d^2z}{dt^2}-z\frac{d^2y}{dt^2} = -\mu\frac{yz}{r^3}+ \mu\frac{yz}{r^3}=0$ .

Since

$y\frac{d^2z}{dt^2}-z\frac{d^2y}{dt^2} = \frac{dy}{dt}\frac{dz}{dt}+y\frac{d^2z}{dt^2}-\frac{dz}{dt}\frac{dy}{dt}-z\frac{d^2y}{dt^2}=\frac{d}{dt}(y\frac{dz}{dt}-z\frac{dy}{dt})$ ,

it must be true that

$\frac{d}{dt}(y\frac{dz}{dt}-z\frac{dy}{dt}) = 0$ .

i.e.

$y\frac{dz}{dt}-z\frac{dy}{dt}=A\quad\quad\quad(0-4)$

where $A$ is a constant.

Similarly,

$z\frac{dx}{dt}-x\frac{dz}{dt}= B,\quad\quad\quad(0-5)$

$x\frac{dy}{dt}-y\frac{dx}{dt}= C\quad\quad\quad(0-6)$

where $B,C$ are constants.

Consequently,

$Ax=xy\frac{dz}{dt} - xz\frac{dy}{dt}$ ,

$By=yz\frac{dx}{dt} - xy\frac{dz}{dt}$ ,

$Cz=xz\frac{dy}{dt}-yz\frac{dx}{dt}$ .

Hence

$Ax + By +Cz=0.\quad\quad\quad(0-7)$

If $C \ne 0$ then by the following well known theorem in Analytic Geometry:

“If A, B, C and D are constants and A, B, and C are not all zero, then the graph of the equation Ax+By+Cz+D=0 is a plane“,

(0-7) represents a plane in the x-y-z coordinate system.

For $C=0$ , we have

$\frac{d}{dt}(\frac{y}{x})=\frac{x\frac{dy}{dt}-y\frac{dx}{dt}}{x^2}\overset{(0-6)}{=}\frac{C}{x^2}=\frac{0}{x^2}=0$ .

It means

$\frac{y}{x}=k$

where $k$ is a constant. Simply put,

$y=k x$ .

Hence, (0-7) still represents a plane in the x-y-z coordinate system (see Fig. 3(a)).

Fig. 3

The implication is that the planet moves around the sun on a plane (see Fig. 4).

Fig. 4

By rotating the axes so that the orbit of the planet is on the x-y plane where $z \equiv 0$ (see Fig. 3), we simplify the equations (0-1)-(0-3) to

$\begin{cases} \frac{d^2x}{dt^2}=-\mu\frac{x}{r^3} \\ \frac{d^2y}{dt^2}=-\mu\frac{y}{r^3}\end{cases}.\quad\quad\quad(1-1)$

It follows that

$\frac{d}{dt}((\frac{dx}{dt})^2 + (\frac{dy}{dt})^2)$

$= 2\frac{dx}{dt}\cdot\frac{d^2x}{dt^2} + 2 \frac{dy}{dt}\cdot\frac{d^2y}{dt^2}$

$\overset{(1-1)}{=}2\frac{dx}{dt}\cdot(-\mu\frac{x}{r^3})+ 2\frac{dy}{dt}\cdot(-\mu\frac{y}{r^3})$

$= -\frac{\mu}{r^3}\cdot(2x\frac{dx}{dt}+2y\frac{dy}{dt})$

$= -\frac{\mu}{r^3}\cdot\frac{d(x^2+y^2)}{dt}$

$= -\frac{\mu}{r^3}\cdot\frac{dr^2}{dt}$

$= -\frac{\mu}{r^3} \cdot 2r \cdot \frac{dr}{dt}$

$= -\frac{2\mu}{r^2} \cdot \frac{dr}{dt}$ .

i.e.,

$\frac{d}{dt}((\frac{dx}{dt})^2 + (\frac{dy}{dt})^2) = -\frac{2\mu}{r^2} \cdot \frac{dr}{dt}$ .

Integrate with respect to $t$ ,

$(\frac{dx}{dt})^2+(\frac{dy}{dt})^2 = \frac{2\mu}{r} + c_1\quad\quad\quad(1-2)$

where $c_1$ is a constant.

We can also re-write (0-6) as

$x\frac{dy}{dt}-y\frac{dx}{dt}=c_2\quad\quad\quad(1-3)$

where $c_2$ is a constant.

Using polar coordinates

$\begin{cases} x= r\cos(\theta) \\ y=r\sin(\theta) \end{cases},$

Fig. 5

we obtain from (1-2) and (1-3) (see Fig. 5):

$(\frac{dr}{dt})^2 + (r\frac{d\theta}{dt})^2-\frac{2\mu}{r} = c_1,\quad\quad\quad(1-4)$

$r^2\frac{d\theta}{dt} = c_2.\quad\quad\quad(1-5)$

If the speed of planet at time $t$ is $v$ then from Fig. 6,

Fig. 6

$v = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta l}{\Delta t} = \lim\limits_{\Delta t\rightarrow 0}\frac{l_r}{\Delta t}\overset{l_r=r\Delta \theta}{=}\lim\limits_{\Delta t \rightarrow 0}\frac{r\cdot \Delta \theta}{\Delta t}=r\cdot\lim\limits_{\Delta t\rightarrow 0}\frac{\Delta \theta}{\Delta t}=r\cdot\frac{d\theta}{dt}$

gives

$v = r\frac{d\theta}{dt}.\quad\quad\quad(1-6)$

Suppose at $t=0$ , the planet is at the greatest distance from the sun with $r=r_0, \theta=0$ and speed $v_0$ . Then the fact that $r$ attains maximum at $t=0$ implies $(\frac{dr}{dt})_{t=0}=0$ . Therefore, by (1-4) and (1-5),

$(\frac{dr}{dt})^2_{t=0} + (r\frac{d\theta}{dt})^2_{t=0}-\frac{s\mu}{r} = 0+ v_0^2-\frac{2\mu}{r}=c_1,$

$r (r\frac{d\theta}{dt})_{t=0}=r_0v_0=c_2.$

i.e.,

$c_1=v_0^2-\frac{2\mu}{r_0},\quad\quad\quad(1-7)$

$c_2=v_0 r_0.\quad\quad\quad(1-8)$

We can now express (1-4) and (1-5) as:

$\frac{dr}{dt} = \pm \sqrt{c_1+\frac{2\mu}{r}-\frac{c_2^2}{r^2}},\quad\quad\quad(1-9)$

$\frac{d\theta}{dt} = \frac{c_2}{r^2}.\quad\quad\quad(1-10)$

Let

$\rho = \frac{c_2}{r}\quad\quad\quad(1-11)$

then

$\frac{d\rho}{dr} = -\frac{c_2}{r^2},\quad\quad\quad(1-12)$

$r=\frac{c_2}{\rho}.\quad\quad\quad(1-13)$

By chain rule,

$\frac{d\theta}{dt} = \frac{d\theta}{d\rho}\cdot\frac{d\rho}{dr}\cdot\frac{dr}{dt}$ .

Thus,

$\frac{d\theta}{d\rho} = \frac{\frac{d\theta}{dt}}{ \frac{d\rho}{dr} \cdot \frac{dr}{dt}}$

$\overset{(1-10), (1-12), (1-9)}{=} \frac{\frac{c_2}{r^2}}{ (-\frac{c_2}{r^2})\cdot(\pm\sqrt{c_1+\frac{2\mu}{r}-\frac{c_2^2}{r^2}}) }$

$\overset{(1-11)}{=} \mp\frac{1}{\sqrt{c_1-\rho^2+2\mu(\frac{\rho}{c_2})}}$

$= \mp\frac{1}{\sqrt{c_1+(\frac{\mu}{c_1})^2-\rho^2+2\mu(\frac{\rho}{c_2}) -(\frac{\mu}{c_2})^2}}$

$= \mp\frac{1}{\sqrt{c_1+(\frac{\mu}{c_1})^2-(\rho^2-2\mu(\frac{\rho}{c_2}) +(\frac{\mu}{c_2})^2)}}$

$= \mp\frac{1}{\sqrt{c_1+(\frac{\mu}{c_1})^2-(\rho-\frac{\mu}{c_2})^2}}$ .

That is,

$\frac{d\theta}{d\rho} = \mp\frac{1}{\sqrt{c_1+(\frac{\mu}{c_1})^2-(\rho-\frac{\mu}{c_2})^2}}.\quad\quad\quad(1-14)$

Since

$c_1+(\frac{\mu}{c_2})^2\overset{(1-7)}{=}v_0^2-\frac{2\mu}{r_0}+(\frac{\mu}{v_0r_0})^2=(v_0-\frac{\mu}{v_0r_0})^2$ ,

we let

$\lambda = \sqrt{c_1 + (\frac{\mu}{c_2})^2}=\sqrt{(v_0-\frac{\mu}{v_0r_0})^2}=|v_0-\frac{\mu}{v_0r_0}|$ .

Notice that $\lambda \ge 0$ .

By doing so, (1-14) can be expressed as

$\frac{d\theta}{d\rho} =\mp \frac{1}{\sqrt{\lambda^2-(\rho-\frac{\mu}{c_2})^2}}$ .

Take the first case,

$\frac{d\theta}{d\rho} = -\frac{1}{\sqrt{\lambda^2-(\rho-\frac{\mu}{c_2})^2}}$ .

Integrate it with respect to $\rho$ gives

$\theta + c = \arccos(\frac{\rho-\frac{\mu}{c_2}}{\lambda})$

where $c$ is a constant.

When $r=r_0, \theta=0$ ,

$c = \arccos(1)=0$ or $c = \arccos(-1) = \pi$ .

And so,

$\theta = \arccos(\frac{\rho-\frac{\mu}{c_2}}{\lambda})$ or $\theta+\pi = \arccos(\frac{\rho-\frac{\mu}{c_2}}{\lambda})$ .

For $c = 0$ ,

$\lambda\cos(\theta) = \rho-\frac{\mu}{c_2}$ .

By (1-11), it is

$\frac{c_2}{r}-\frac{\mu}{c_2} = \lambda \cos(\theta).\quad\quad\quad(1-15)$

Fig. 7

Solving (1-15) for $r$ yields

$r=\frac{c_2^2}{c_2 \lambda \cos(\theta)+\mu}=\frac{\frac{c_2^2}{\mu}}{\frac{c_2}{\mu}\lambda \cos(\theta)+1}\overset{p=\frac{c_2^2}{\mu}, e=\frac{c_2 \lambda}{\mu}}{=}\frac{p}{e \cos(\theta) + 1}$ .

i.e.,

$r = \frac{p}{e \cos(\theta) + 1}.\quad\quad\quad(1-16)$

Studies in Analytic Geometry show that for an orbit expressed by (1-16), there are four cases to consider depend on the value of $e$ :

We can rule out parabolic and hyperbolic orbit immediately for they are not periodic. Given the fact that a circle is a special case of an ellipse, it is fair to say:

The orbit of a planet is an ellipse with the Sun at one of the two foci.

In fact, this is what Kepler stated as his first law of planetary motion.

Fig. 8

For $c=\pi$ ,

$\theta + \pi = \arccos(\frac{\rho-\frac{\mu}{c_2}}{\lambda})$

from which we obtain

$r=\frac{c_2^2}{c_2 \lambda \cos(\theta+\pi)+\mu}=\frac{\frac{c_2^2}{\mu}}{\frac{c_2}{\mu}\lambda\cos(\theta+\pi)+1}\overset{p=\frac{c_2^2}{\mu}, e=\frac{c_2 \lambda}{\mu}}{=}\frac{p}{e \cos(\theta+\pi) + 1}.\quad\quad(1-17)$

This is an ellipse. Namely, the result of rotating (1-16) by hundred eighty degrees or assuming $r$ attains its minimum at $t=0$ .

The second case

$\frac{d\theta}{d\rho} = +\frac{1}{\sqrt{\lambda^2-(\rho-\frac{\mu}{c_2})^2}}$

can be written as

$-\frac{d\theta}{d\rho} = -\frac{1}{\sqrt{\lambda^2-(\rho-\frac{\mu}{c_2})^2}}$ .

Integrate it with respect to $\rho$ yields

$-\theta + c = \arccos(\frac{\rho-\frac{\mu}{c_2}}{\lambda})$

from which we can obtain (1-16) and (1-17) again.

Fig. 9

Over the time duration $\Delta t$ , the area a line joining the sun and a planet sweeps an area $A$ (see Fig. 9).

$A = \int\limits_{t}^{t+\Delta t}\frac{1}{2}r\cdot v\;dt \overset{(1-6)}{=} \int\limits_{t}^{t+\Delta t}\frac{1}{2}r\cdot r\frac{d\theta}{dt}\;dt=\int\limits_{t}^{t+\Delta t}\frac{1}{2}r^2\frac{d\theta}{dt}\;dt\overset{(1-5)}{=}\int\limits_{t}^{t+\Delta t}\frac{1}{2}c_2\;dt = \frac{1}{2}c_2\Delta t$ .

It means

$A = \frac{1}{2}c_2\Delta t\quad\quad\quad(2-1)$

or that

$\frac{A}{\Delta t} = \frac{1}{2}c_2$

is a constant. Therefore,

A line joining the Sun and a planet sweeps out equal areas during equal intervals of time.

This is Kepler’s second law. It suggests that the speed of the planet increases as it nears the sun and decreases as it recedes from the sun (see Fig. 10).

Fig. 10

Furthermore, over the interval $T$ , the period of the planet’s revolution around the sun, the line joining the sun and the planet sweeps the enire interior of the planet’s elliptical orbit with semi-major axis $a$ and semi-minor axis $b$ . Since the area enlosed by such orbit is $\pi ab$ (see “Evaluate a Definite Integral without FTC“), setting $\Delta t$ in (2-1) to $T$ gives

${\frac{1}{2}c_2 T} = {\pi a b} \implies {\frac{1}{4}c_2^2 T^2}={\pi^2 a^2 b^2} \implies T^2 = \frac{4\pi^2 a^2 b^2}{c_2^2} \implies \frac{T^2}{a^3} = \frac{4\pi^2b^2}{c_2^2a}. (3-1)$

While we have $p = \frac{c_2^2}{\mu}$ in (1-16), it is also true that for such ellipse, $p=\frac{b^2}{a}$ (see “An Ellipse in Its Polar Form“). Hence,

$\frac{b^2}{a}=\frac{c_2^2}{\mu}\implies c_2^2=\frac{\mu b^2}{a}.\quad\quad\quad(3-2)$

Substituting (3-2) for $c_2^2$ in (3-1),

$\frac{T^2}{a^3} = \frac{4\pi^2 b^2}{(\frac{\mu b^2}{a})a}=\frac{4\pi^2}{\mu} \overset{\mu=GM}{=}\frac{4\pi^2}{GM}.\quad\quad\quad(3-3)$

Thus emerges Kepler’s third law of planetary motion:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Established by (3-3) is the fact that the proportionality constant is the same for all planets orbiting around the sun.

This image has an empty alt attribute; its file name is main-qimg-e7446d7decf4bdc83f8b712109470213.jpg

Déjà vu!

April 17, 2019April 24, 2019 / Michael Xue / Leave a comment

The stages of a two stage rocket have initial masses $m_1$ and $m_2$ respectively and carry a payload of mass $P$ . Both stages have equal structure factors and equal relative exhaust speeds. If the rocket mass, $m_1+m_2$ , is fixed, show that the condition for maximal final speed is

$m_2^2 + P m_2 = P m_1$ .

Find the optimal ratio $\frac{m_1}{m_2}$ when $\frac{P}{m_1+m_2} = b$ .

According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two stage rocket is

$v = -c \log(1-\frac{e \cdot m_1}{m_1 + m_2 +P})- c\log(1-\frac{e \cdot m_2}{m_2+P})$

Let $m_0 = m_1 + m_2$ , we have

$m_1 = m_0-m_2$

and,

$v = -c \log(1-\frac{e \cdot (m_0-m_2)}{m_0 +P})- c\log(1-\frac{e \cdot m_2}{m_2+P})$

Differentiate $v$ with respect to $m_2$ gives

$v' = \frac{c(e-1)e(2m_2 P-m_0 P +m_2^2)}{(P+m_2)(P-e m_2+m_2)(P+e m_2-e m_0+m_0)}= \frac{c(e-1)e(2m_2-m_0 P+m_2^2)}{(P+m_2)(P+(1-e)m_2)(P+e m_2 + (1-e)m_0)}\quad\quad\quad(1)$

It follows that $v'=0$ implies

$2m_2 P-m_0 P + m_2^2 =0$ .

That is, $2 m_2 P - (m_1+m_2) P + m_2^2 = (m_2-m_1) P + m_2^2=0$ . i.e.,

$m_2^2 + P m_2 = P m_1\quad\quad\quad(2)$

It is the condition for an extreme value of $v$ . Specifically, the condition to attain a maximum (see Exercise-2)

When $\frac{P}{m_1+m_2} = b$ , solving

$\begin{cases} (m_2-m_1)P+m_2^2=0\\ \frac{P}{m_1+m_2} = b\end{cases}$

yields two pairs:

$\begin{cases} m_1=\frac{(\sqrt{b}\sqrt{b+1}+b+1)P}{b}\\m_2= -\frac{\sqrt{b^2+b}P+bP}{b}\end{cases}$

and

$\begin{cases} m_1= - \frac{(\sqrt{b}\sqrt{b+1}-b-1)P}{b}\\ m_2=\frac{\sqrt{b^2+b}P-bP}{b}\end{cases}\quad\quad\quad(3)$

Only (3) is valid (see Exercise-1)

Hence

$\frac{m_1}{m_2} = -\frac{(\sqrt{b}\sqrt{b+1}-b-1)P}{\sqrt{b^2+b}P-b P} = \frac{\sqrt{b+1}}{\sqrt{b}} = \sqrt{1+\frac{1}{b}}$

The entire process is captured in Fig. 2.

Fig. 2

Exercise-1 Given $b>0, 0<e<1, P>0$ , prove:

$- \frac{(\sqrt{b}\sqrt{b+1}-b-1)P}{b}>0$
$\frac{\sqrt{b^2+b}P-bP}{b}>0$

Exercise-2 From (1), prove the extreme value attained under (2) is a maximum.

Boosting rocket flight performance without calculus

April 8, 2019June 12, 2022 / Michael Xue / Leave a comment

Fig. 1

The stages of a two-stage rocket have initial masses $m_1$ and $m_2$ respectively and carry a payload of mass $P$ . Both stages have equal structure factors $e$ and equal relative exhaust speed $c$ . The rocket mass, $m_1+m_2$ is fixed and $\frac{P}{m_1+m_2}=b$ .

According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two-stage rocket is

$v = -c \log(1-\frac{em_1}{m_1+m_2+P}) - c\log(1-\frac{em_2}{m_2+P})$

Let $a = \frac{m1}{m_2}$ , it becomes

$v = -c\log(1-\frac{ea}{a+1+b(a+1)})-c \log(1-\frac{e}{1+b(a+1)})$

where $a>0, b>0, c>0, 0< e < 1$ . We will maximize $v$ with an appropriate choice of $a$ .

That is, given

$v = c\log(\frac{(a+1)(b+1)((a+1)b+1)}{(1-e+(a+1)b)((1-e)a+(a+1)b+1})$

where $a>0, b>0, c>0, 0<e<1$ . Maximize $v$ with an appropriate value of $a$ .

The above optimization problem is solved using calculus (see “Viva Rocketry! Part 2“). However, there is an alternative that requires only high school mathematics with the help of a Computer Algebra System (CAS). This non-calculus approach places more emphasis on problem solving through mathematical thinking, as all symbolic calculations are carried out by the CAS (e.g., see Fig. 2). It also makes a range of interesting problems readily tackled with minimum mathematical prerequisites.

The fact that

$\log$ is a monotonic increasing function $\implies v_{max} = c\log(w_{max})$

where

$w = \frac{(a+1)(b+1)((a+1)b+1)}{(1-e+(a+1)b)((1-e)a+(a+1)b+1)}$

$w(1-e+(a+1)b)((1-e)a+(a+1)b+1) - (a+1)(b+1)((a+1)b+1)=0\quad\quad\quad(1)$

(1) can be written as

$A_1 a^2 + B_1 a +C_1= 0$

where

$A_1 = -bew+b^2w+bw-b^2-b$

$B_1 = e^2w-2bew-2ew+2b^2w+3bw+w-2b^2-3b-1$ ,

$C_1 = -bew-ew+b^2w+2bw+w-b^2-2b-1$ .

Since $A_1 = 0$ means

$-b e w + b^2 w + b w - b^2 - b =0$ .

That is

$w = -\frac{b+1}{e-b-1}$ .

Solve

$\frac{(a+1)(b+1)((a+1)b+1)}{(1-e+(a+1)b)((1-e)a+(a+1)b+1)} = -\frac{b+1}{e-b-1}$

for $a$ gives $a = 0 \implies A_1 \neq 0$ if $a > 0$ .

Hence, (1) is a quadratic equation. For it to have solution, its discriminant $B_1^2-4A_1C_1$ must be nonnegative, i.e.,

$(e^2-2be-2e+b+1)^2 w^2-2(b+1)(2be^2+e^2-2be-2e+b+1) w +(b+1)^2 \geq 0\quad(2)$

Consider

$(e^2-2be-2e+b+1)^2 w^2-2(b+1)(2be^2+e^2-2be-2e+b+1)w+(b+1)^2 = 0\quad(3)$

If $e^2-2be-2e+b+1 \neq 0$ , (3) is a quadratic equation.

Solving (3) yields two solutions

$w_1 = -\frac{(b+1)(2\sqrt{b(b+1)}e^2-2be^2-e^2-2\sqrt{b(b+1)}e+2be+2e-b-1)}{(e^2-2be-2e+b+1)^2}$ ,

$w_2=\frac{(b+1)(2\sqrt{b(b+1)}e^2+2be^2+e^2-2\sqrt{b(b+1)}e-2be-2e+b+1)}{(e^2-2be-2e+b+1)^2}$ .

Since $0 < e < 1$ ,

$w_1 - w_2 = -\frac{4(b+1)\sqrt{b(b+1)}(e-1)e}{(e^2-2be-2e+b+1)^2} > 0\quad(4)$

(4) implies

$w_1 > w_2$

and, the solution to (2) is

$w \leq w_2$ or $w \ge w_1$

i.e.,

$w \leq \frac{(b+1)(2\sqrt{b(b+1)}e^2+2be^2+e^2-2\sqrt{b(b+1)}e-2be-2e+b+1)}{(e^2-2be-2e+b+1)^2}\quad\quad\quad(4)$

$w \ge -\frac{(b+1)(2\sqrt{b(b+1)}e^2-2be^2-e^2-2\sqrt{b(b+1)}e+2be+2e-b-1)}{(e^2-2be-2e+b+1)^2}\quad\quad\quad(5)$

We prove that (4) is true by showing (5) is false:

Consider $w - w_1= 0$ :

$\frac{(b+1)(e-1) e \cdot f(a)}{(1-e+ab+b)(a(1-e)+ab+b+1)(e^2-2be-2e+b+1)^2} = 0\quad\quad\quad(6)$

where

$f(a) = 2a\sqrt{b(b+1)}e^2+a^2be^2+be^2+e^2-2a^2b\sqrt{b(b+1)}$

$-4ab\sqrt{b(b+1)}e - 2b\sqrt{b(b+1)}e - 4a\sqrt{b(b+1)}e$

$-2\sqrt{b(b+1)}e-2a^2b^2e-4ab^2e-2a^2be-4abe-4be$

$-2e+2a^2b^2\sqrt{b(b+1)}+4ab^2\sqrt{b(b+1)} + 2b^2\sqrt{b(b+1)} +2a^2b\sqrt{b(b+1)}$

$+6ab\sqrt{b(b+1)} + 4b\sqrt{b(b+1)} + 2a\sqrt{b(b+1)} + 2\sqrt{b(b+1)}$

$+2a^2b^3 + 4ab^3 + 2b^3 +3a^2b^2 + 8ab^2 +5b^2+a^2b+4ab+4b+1$ .

It can be written as

$A_2a^2 + B_2a + C_2\quad\quad\quad(7)$

where

$A_2 = be^2-2b\sqrt{b(b+1)}e-2b^2e-2be+2b^2\sqrt{b(b+1)}+2b\sqrt{b(b+1)}$

$+2b^3+3b^2+b$ ,

$B_2 = 2\sqrt{b(b+1)}e^2-4b\sqrt{b(b+1)}e -4\sqrt{b(b+1)}e$

$-4b^2e-4be+4b^2\sqrt{b(b+1)}+6b\sqrt{b(b+1)}+2\sqrt{b(b+1)}+4b^3+8b^2+4b$ ,

$C_2=be^2+e^2-2b\sqrt{b(b+1)}e-2\sqrt{b(b+1)}e-2b^2e-4be$

$-2e+2b^2\sqrt{b(b+1)}+4b\sqrt{b(b+1)}+2\sqrt{b(b+1)}+2b^3+5b^2+4b+1$ .

Since $A_2 > 0$ (see Exercise 1) and,

solve (7) for $a$ yields

$a = -\sqrt{1+\frac{1}{b}}$ .

It follows that for $a > 0, f(a) > 0$ .

Consequently, $w-w_1$ is a negative quantity. i.e.,

$w-w_1 < 0$

which tells that (5) is false.

Hence, when $e^2-2be-2e+b+1 \neq 0$ , the global maximum $w_{max}$ is $w_2$ .

Solving $w = w_2$ for $a$ :

$\frac{(a+1)(b+1)((a+1)b+1}{(1-e+(a+1)b)((1-e)a+(a+1)b+1)} = \frac{(b+1)(2\sqrt{b(b+1)}e^2+2be^2+e^2-2\sqrt{b(b+1)}e-2be-2e+b+1)}{(e^2-2be-2e+b+1)^2}$ ,

we have

$a = \sqrt{1+ \frac{1}{b}}$ .

Therefore,

$e^2-2be-2e+b+1 \neq 0 \implies w$ attains maximum at $a = \sqrt{1+ \frac{1}{b}}$ .

In fact, $w$ attains maxima at $a = \sqrt{1+\frac{1}{b}}$ even when $e^2-2be-2e+b+1 = 0$ , as shown below:

Solving $e^2-2be-2e+b+1 = 0$ for $e$ , we have

$e_1= -\sqrt{b(b+1)}+b+1$ or $e_2 = \sqrt{b(b+1)} + b + 1$ .

Only $e_1$ is valid (see Exercise-2),

When $e = e_1$ ,

$w(\sqrt{1+\frac{1}{b}} )- w(a) = - \frac{(b+1)g(a)}{4 \sqrt{b(b+1)} (\sqrt{b(b+1)}+ab) (a\sqrt{b(b+1)}+b+1)}\quad(8)$

where

$g(a) = (2a^2b+4ab+2b+2a+2)\sqrt{b(b+1)}-2a^2b^2-4ab^2-2b^2-a^2b-4ab-3b-1$

Solve quadratic equation $g(a) = 0$ for $a$ yields

$a = \sqrt{1+\frac{1}{b}}$ .

The coefficient of $a^2$ in $g(a)$ is $2b\sqrt{b(b+1)}-2b^2-b$ , a negative quantity (see Exercise-3).

The implication is that $g(a)$ is a negative quantity when $a \neq \sqrt{1 + \frac{1}{b}}$ .

Hence, (8) is a positive quantity, i.e.,

$e^2-2be-2e+b+1 = 0, a \neq \sqrt{1+\frac{1}{b}} \implies w(\sqrt{1+\frac{1}{b}})-w(a) > 0$

We therefore conclude

$\forall 0 < e < 1, b > 0, w$ attains its maximum at $a = \sqrt{1+\frac{1}{b}}$ .

Fig. 2

Exercise-1 Prove: $0<e<1, b>0 \implies$

$be^2-2b\sqrt{b(b+1)}e-2b^2e-2be+2b^2\sqrt{b(b+1)}+2b\sqrt{b(b+1)}+2b^3+3b^2+b > 0$

Exercise-2 Prove: $b > 0 \implies 0 <-\sqrt{b(b+1)} + b +1 <1$

Exercise-3 Prove: $b > 0 \implies 2b\sqrt{b(b+1)}-2b^2-b < 0$

Viva Rocketry! Part 2

January 31, 2019September 26, 2022 / Michael Xue

Screen Shot 2018-12-24 at 5.13.14 AM.png

Fig. 1

A rocket with $n$ stages is a composition of $n$ single stage rocket (see Fig. 1) Each stage has its own casing, instruments and fuel. The $n$ th stage houses the payload. This image has an empty alt attribute; its file name is screen-shot-2019-01-09-at-6.24.56-pm.png

This image has an empty alt attribute; its file name is screen-shot-2019-01-09-at-6.24.56-pm.png

Fig. 2

The model is illustrated in Fig. 2, the $i^{th}$ stage having initial total mass $m_i$ and containing fuel $\epsilon_i m_i (0 < \epsilon_i <1, 1 \leq i \leq n)$ . The exhaust speed of the $i^{th}$ stage is $c_i$ . The flight of multi-stage rocket starts with the $1^{st}$ stage fires its engine and the rocket is lifted. When all the fuel in the $1^{st}$ stage has been burnt, the $1^{st}$ stage’s casing and instruments are detached. The remaining stages of the rocket continue the flight with $2^{nd}$ stage’s engine ignited. Generally, the rocket starts its $i^{th}$ stage of flight with final velocity achieved at the end of previous stage of flight. The entire rocket is propelled by the fuel in the $i^{th}$ casing of the rocket. When all the fuel for this stage has been burnt, the $i^{th}$ casing is separated from the rest of the stages. The flight of the rocket is completed if $i=n$ . Otherwise, it enters the next stage of flight. When all external forces are omitted, the governing equation of rocket’s $i^{th}$ stage flight (see “Viva Rocketry! Part 1” or “An alternate derivation of ideal rocket’s flight equation“) is

$0 = m_i(t) \frac{dv_i(t)}{dt} + c_i \frac{dm_i(t)}{dt}$

It can be written as

$\frac{dv_i(t)}{dt} = -\frac{c_i}{m_i(t)}\frac{dm_i(t)}{dt}\quad\quad\quad(1)$

Integrate (1) from $t_0$ to $t$ ,

$\int\limits_{t_0}^{t}\frac{dv_i(t)}{dt}\;dt = -c_i \int\limits_{t_0}^{t}\frac{1}{m_i(t)}\frac{dm_(t)}{dt}\;dt$

gives

$v_i(t) - v_i(t_0) = -c_i(\log(m_i(t))-\log(m_i(t_0)))=-c_i\log(\frac{m_i(t)}{m_i(t_0)})$ .

At $t=t^*$ when $i^{th}$ stage’s fuel has been burnt, we have

$v_i(t^*) - v_i(t_0) = -c_i\log(\frac{m_i(t^*)}{m_i(t_0)})\quad\quad\quad(2)$

where

$m_i(t_0) = \sum\limits_{k=i}^{n} m_k +P$

and,

$m_i(t^*) = \sum\limits_{k=i}^n m_k - \epsilon_i m_i + P$ .

Let $v^*_i = v_i(t^*), \; v^*_{i-1}$ the velocity of rocket at the end of ${i-1}^{th}$ stage of flight. Since $v_i(t_0) = v^*_{i-1}$ , (2) becomes

$v^*_i - v^*_{i-1} = -c_i \log(\frac{ \sum\limits_{k=i}^n m_k - \epsilon_i m_i+P}{\sum\limits_{k = i}^{n} m_k + P}) = -c_i\log(1-\frac{\epsilon_i m_i}{\sum\limits_{k = i}^{n} m_k + P})$

i.e.,

$v^*_i = v^*_{i-1}-c_i\log(1-\frac{\epsilon_i m_i}{\sum\limits_{k = i}^{n} m_k + P}), \quad\quad 1\leq i \leq n, v_0=0\quad\quad\quad(3)$

For a single stage rocket ( $n=1$ ), (3) is

$v_1^*=-c_1\log(1-\frac{\epsilon_1}{1+\beta})\quad\quad\quad(4)$

In my previous post “Viva Rocketry! Part 1“, it shows that given $c_1=3.0\;km\;s^{-1}, \epsilon_1=0.8$ and $\beta=\frac{1}{100}$ , (4) yields $4.7 \;km\;s^{-1}$ , a value far below $7.8\;km\;s^{-1}$ , the required speed of an earth orbiting satellite.

But is there a value of $\beta$ that will enable the single stage rocket to produce the speed a satellite needs? Let’s find out. Differentiate (4) with respect to $\beta$ gives

$\frac{dv_1^*}{d\beta} = - \frac{c_1 \epsilon_1}{(\beta+1)^2 (1-\frac{\epsilon_1}{\beta+1})} = - \frac{c_1\epsilon_1}{(\beta+1)^2(\frac{1-\epsilon_1+\beta}{\beta+1})} < 0$

since $c_1, \beta$ are positive quantities and $0< \epsilon_1 < 1$ . It means $v_1^*$ is a monotonically decreasing function of $\beta$ . Moreover,

$\lim\limits_{\beta \rightarrow 0}v_1^*=\lim\limits_{\beta \rightarrow 0}-c_1\log(1-\frac{\epsilon_1}{1+\beta}) = - c_1 \log(1-\epsilon_1)\quad\quad\quad(5)$

Given $c_1=3.0\;km\;s^{-1}, \epsilon_1=0.8$ , (5) yields approximately

$4.8\;km\;s^{-1}$

This image has an empty alt attribute; its file name is screen-shot-2019-01-21-at-4.56.14-pm.png

Fig. 3

This upper limit implies that for the given $c_1$ and $\epsilon_1$ , no value of $\beta$ will produce a speed beyond (see Fig. 4)

Let’s now turn to a two stage rocket ( $n=2$ ) From (3), we have

$v_2^* = -c_1\log(1-\frac{\epsilon_1 m_1}{m_1+m_2+P}) - c_2\log(1-\frac{\epsilon_2 m_2}{m_2+P})\quad\quad\quad(6)$

If $c_1=c_2=c, \epsilon_1 = \epsilon_2 = \epsilon, m_1=m_2$ and $\frac{P}{m_1+m_2} = \beta$ , then

$P = \beta (m_1+m_2) = 2m_1\beta = 2m_2\beta$ .

Consequently,

$v_2^*=-c \log(1-\frac{\epsilon}{2(1+\beta)}) - c\log(1-\frac{\epsilon}{1+2\beta})\quad\quad\quad(7)$

When $c=3.0\;km\;s^{-1}, \epsilon=0.8$ and $\beta = \frac{1}{100}$ ,

$v_2^* \approx 6.1\;km\;s^{-1}$

This image has an empty alt attribute; its file name is screen-shot-2019-01-21-at-10.30.21-am.png

Fig. 5

This is a considerable improvement over the single stage rocket ( $v^*=4.7\; km\;s^{-1}$ ). Nevertheless, it is still short of producing the orbiting speed a satellite needs. In fact,

$\frac{dv_2^*}{d\beta} = -\frac{2c\epsilon}{(2\beta+2)^2(1-\frac{\epsilon}{2\epsilon+2})}-\frac{2c\epsilon}{(2\beta+1)^2(1-\frac{\epsilon}{2\beta+1})}= -\frac{2c\epsilon}{(2\beta+2)^2\frac{2\beta+2-\epsilon}{2\beta+2}}-\frac{2c\epsilon}{(2\beta+1)^2\frac{2\beta+1-\epsilon}{2\beta+1}} < 0$

indicates that $v_2^*$ is a monotonically decreasing function of $\beta$ . In addition,

$\lim\limits_{\beta \rightarrow 0} v_2^*=\lim\limits_{\beta \rightarrow 0 }-c\log(1-\frac{\epsilon}{2+2\beta})-c\log(1-\frac{\epsilon}{1+2\beta})=-c\log(1-\frac{\epsilon}{2})-c\log(1-\epsilon)$ .

Therefore, there is an upper limit to the speed a two stage rocket can produce. When $c=3.0\;km\;s^{-1}, \epsilon=0.8$ , the limit is approximately

$6.4\;km\;s^{-1}$

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Fig. 6

In the value used above, we have taken equal stage masses, $m_1 = m_2$ . i.e., the ratio of $m_1 : m_2 = 1 : 1$ . Is there a better choice for the ratio of $m_1:m_2$ such that a better $v_2^*$ can be obtained? To answer this question, let $\frac{m_1}{m_2} = \alpha$ , we have

$m_1 = \alpha m_2\quad\quad\quad(8)$

Since $P = \beta (m_1+m_2)$ , by (8),

$P = \beta (\alpha m_2 + m_2)\quad\quad\quad(9)$

Substituting (8), (9) into (6), $v_2^* = -c\log(1-\frac{\epsilon \alpha m_2}{\alpha m_2+m_2+\beta(\alpha m_2+m_2))}) - c\log(1-\frac{\epsilon m_2}{m_2 + \beta(\alpha m_2 + m_2)})$ $= -c\log(1-\frac{\epsilon \alpha}{\alpha+1+\beta(\alpha+1)})-c\log(1-\frac{\epsilon}{1+\beta(\alpha+1)})$ $= c\log\frac{(\alpha+1)(\beta+1)((\alpha+1)\beta+1)}{(1-\epsilon+(\alpha+1)\beta)((1-\epsilon)\alpha + (\alpha+1)\beta+1)}\quad\quad\quad(10)$ a function of $\alpha$ . It can be written as

$v_2^*(\alpha) = c\log(w(\alpha))$

where

$w(\alpha) = \frac{(\alpha+1)(\beta+1)((\alpha+1)\beta+1)}{(1-\epsilon+(\alpha+1)\beta)((1-\epsilon)\alpha + (\alpha+1)\beta+1)}$

This is a composite function of $\log$ and $w$ . Since $\log$ is a monotonic increasing function (see “Introducing Lady L“),

$(v_2^*)_{max} = c\log(w_{max})$

Here, $(v_2^*)_{max}, w_{max}$ denote the maximum of $v_2^*$ and $w$ respectively. To find $w_{max}$ , we differentiate $w$ ,

$\frac{dw}{d\alpha} = \frac{(\beta+1)(\alpha^2\beta-\beta-1)(\epsilon-1)\epsilon}{(\epsilon-\alpha \beta-\beta-1)^2(\alpha\epsilon-\alpha \beta-\beta-\alpha-1)^2}=\frac{(\beta+1)(\alpha^2\beta-\beta-1)(\epsilon-1)\epsilon}{(1-\epsilon+\alpha \beta+\beta)^2(\alpha(1-\epsilon)+\alpha \beta+\beta)^2}\quad\quad\quad(11)$

Solving $\frac{dw}{d\alpha} = 0$ for $\alpha$ gives

$\alpha = - \sqrt{1+\frac{1}{\beta}}$ or $\sqrt{1+\frac{1}{\beta}}$ .

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Fig. 7

By (8), the valid solution is

$\alpha = \sqrt{1+\frac{1}{\beta}}$ .

It shows that $w$ attains an extreme value at $\sqrt{1+\frac{1}{\beta}}$ . Moreover, we observe from (11) that

$\alpha < \sqrt{1+\frac{1}{\beta}} \implies \frac{dw}{d\alpha} > 0$

and

$\alpha > \sqrt{1+\frac{1}{\beta}} \implies \frac{dw}{d\alpha} < 0$ .

i.e., $w$ attains maximum at $\alpha=\sqrt{1+\frac{1}{\beta}}$ . It follows that

$v_2^*$ attains maximum at $\alpha = \sqrt{1+\frac{1}{\beta}}$ .

Therefore, to maximize the final speed given to the satellite, we must choose the ratio

$\frac{m_1}{m_2} = \sqrt{1+\frac{1}{\beta}}$ .

With $\beta =\frac{1}{100}$ , the optimum ratio $\frac{m_1}{m_2}=10.05$ , showing that the first stage must be about ten times large than the second. Using this ratio and keep $\epsilon=0.8, c=3.0\;km\;s^{-1}$ as before, (10) now gives

$v_2 = 7.65\;km\;s^{-1}$ ,

a value very close to the required one. This image has an empty alt attribute; its file name is screen-shot-2019-01-23-at-4.14.53-pm.png

Fig. 8

Setting $\beta = \frac{1}{128}$ , we reach the goal:

$v_2^* = 7.8\;km\;s^{-1}$

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Fig. 9

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Fig. 10

At last, it is shown mathematically that provided the stage mass ratios ( $\frac{P}{m_1+m_2}$ and $\frac{m_1}{m_2}$ )are suitably chosen, a two stage rocket can indeed launch satellites into earth orbit.

Exercise 1. Show that $\alpha < \sqrt{1+\frac{1}{\beta}} \implies \frac{dw}{d\alpha} > 0$ and $\alpha > \sqrt{1+\frac{1}{\beta}} \implies \frac{dw}{d\alpha} < 0$ . Exercise 2. Using the optimum $\frac{m_1}{m_2} = \sqrt{1+\frac{1}{\beta}}$ and $\epsilon=0.8, c=3.0\;km\;s^{-1}$ , solving (10) numerically for $\beta$ such that $v_2^* = 7.8$ .

When rocket ejects its propellant at a variable rate

December 16, 2018August 2, 2021 / Michael Xue / Leave a comment

A rocket is programmed to burn and ejects its propellant at the variable rate $\alpha \cdot k \cdot e^{-kt}$ , where $k$ and $\alpha$ are positive constants. The rocket is launched vertically from rest. Neglecting all external forces except gravity, show that the final speed given to the payload, of mass $P$ , when all the fuel has been burnt is

$v= -c \log(1-{\frac{\epsilon m_0}{m_0+P}}) + {\frac{g}{k}\log(1-{\frac{\epsilon m_0}{\alpha}}})$ .

Here $c$ is the speed of the propellant relative to the rocket, $m_0$ the initial rocket mass, excluding the payload. The initial fuel mass is $\epsilon m_0$ .

From my previous post “An alternative derivation of rocket’s flight equation“, we know in our present context,

$\frac{dm}{dt} = - \; \alpha k e^{-kt}\quad\quad\quad(1)$

Integrate (1) from $0$ to $t$ ,

$m(t)-m(0) = \int\limits_{0}^{t}{ -\alpha k e^{-kt}} dt = \alpha e^{-kt}\bigg|_{0}^{t}=\alpha e^{-kt}-\alpha$

Since $m(0) = m_0 + P$ ,

$m(t) = m_0 + P -\alpha +\alpha e^{-kt}$ .

The rocket’s flight equation now is

$-mg = m \frac{dv}{dt} + c\cdot (-\alpha k e^{-kt})$

i.e.,

$\frac{dv}{dt} = \frac{c\alpha k e^{-kt}}{m_0+P-\alpha+\alpha e^{-kt}} -g\quad\quad\quad(2)$

When all the fuel has been burnt at time $t^*$ ,

$m(t^*) = (1-\epsilon) m_0 + P$ .

That is:

$m_0 + P - \alpha + \alpha e^{-kt^*} = (1-\epsilon) m_0 + P\quad\quad\quad(3)$ .

Solve (3) for $t^*$ ,

$t^* = -\frac{1}{k}\log(1-\frac{\epsilon m_0}{\alpha})$

Integrate (2) from $0$ to $t^*$ , we have

$v(t^*) - v(0) = -c \log(m_0+P-\alpha + \alpha e^{-kt}) \bigg|_{0}^{t^*}- gt\bigg|_{0}^{t^*}$

Since $v(0) = 0$ ,

$v(t^*) = -c \log(\frac{m_0 + P -\alpha + \alpha e^{(-k) \cdot ({-\frac{1}{k}\log(1-\frac{\epsilon m_0}{\alpha}))}}} {m_0 + P -\alpha + \alpha e^{-k\cdot 0}}) - g\cdot( -\frac{1}{k}\log(1-\frac{\epsilon m_0}{\alpha}) - 0)$

$= -c \log(\frac{m_0+P-\alpha +\alpha (1-\frac{\epsilon m_0}{\alpha})}{m_0+P}) + \frac{g}{k}\log(1-\frac{\epsilon m_0}{\alpha})$

$= -c \log(\frac{m_0+P-{\epsilon m_0})}{m_0+P}) + \frac{g}{k}\log(1-\frac{\epsilon m_0}{\alpha})$

gives the final speed

$v(t^*) = -c \log(1-\frac{\epsilon m_0}{m_0+P}) + \frac{g}{k}\log(1-\frac{\epsilon m_0}{\alpha})$

Exercise 1. Using Omega CAS Explorer, solve $(1), (2), (3)$ for $m(t), v(t), t^*$ respectively.

Exercise 2. Before firing, a single stage rocket has total mass $m_0$ , which comprises the casing, instruments etc, with mass $m_c$ , and the fuel. The fuel is programmed to burn and to be ejected at a variable rate such that the total mass of the rocket $m(t)$ at any time $t$ , during which the fuel is being burnt, is given by

$m(t) = m_0 e^{\frac{-kt}{m_0}}$

where $k$ is a constant.

The rocket is launched vertically from rest. Neglect all external forces except gravity, show that the height $h$ attained at the instant the fuel is fully consumed is

$h = \frac{1}{2}(\frac{m_0}{k} \cdot \log{\frac{m_0}{m_c}})^2(\frac{ck}{m_0}-g)$

$c$ being the exhaust speed relative to the rocket.

An alternative derivation of ideal rocket’s flight equation

December 13, 2018August 3, 2021 / Michael Xue / 2 Comments

USS_Cape_St._George_(CG_71)_fires_a_tomahawk_missile_in_support_of_OIF.jpg

I will derive the ideal rocket’s flight equation differently than what is shown in “Viva Rocketry! Part 1”

Let

$\Delta m$ – the mass of the propellant

$m$ – the mass of the rocket at time $t$

$v$ – the speed of the rocket and $\Delta m$ at time $t$

$u$ – the speed of the ejected propellant, relative to the rocket

$p_1$ – the magnitude of $\Delta m$ ‘s momentum

$p_2$ – the magnitude of rocket’s momentum

For the propellant:

$\Delta p_1 = \Delta m \cdot (\boxed {v +\Delta v -u} ) - \Delta m \cdot v$

$=\Delta m\cdot v + \Delta m\cdot \Delta v -\Delta m\cdot u - \Delta m \cdot v$

$= -\Delta m \cdot u + \Delta m \cdot \Delta v$

where $\boxed {v +\Delta v -u}$ is the speed of $\Delta m$ at $t+\Delta t$ (see “A Thought Experiment on Velocities”)

By Newton’s second law,

$F_1 = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta p_1}{\Delta t}$

$= \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t}$ .

For the rocket:

$\Delta p_2 = (m-\Delta m) \cdot (v +\Delta v) - (m-\Delta m)\cdot v$

$= (m-\Delta m)(v +\Delta v-v)$

$= (m-\Delta m)\cdot {\Delta v}$

$= m \cdot \Delta v - \Delta m \cdot \Delta v$

$F_2 = \lim\limits_{\Delta t \rightarrow 0} \frac{\Delta p_2}{\Delta t}$

$= \lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t}$ .

By Newton’s third law,

$F_2 = -F_1$ .

Therefore,

$\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t} = - \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t}$

That is,

$\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t} + \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t} = 0$ .

It implies

$\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v -\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t} = 0$

or,

$\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v -\Delta m \cdot u}{\Delta t} = 0$ .

Since

$\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v -\Delta m \cdot u }{\Delta t}= \lim\limits_{\Delta t \rightarrow 0} {\frac{m\cdot \Delta v -u \cdot \Delta m}{\Delta t}}=m \cdot \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta v}{\Delta t}-u \cdot \lim\limits_{\Delta t \rightarrow 0} \frac{\Delta m}{\Delta t}$ ,

$\frac{dv}{dt} = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta v}{\Delta t}$ ,

and

$\lim\limits_{\Delta t \rightarrow 0}\frac{\Delta m}{\Delta t}=\lim\limits_{\Delta t \rightarrow 0} \frac{m(t) - m(t+\Delta t)}{\Delta t}= -\lim\limits_{\Delta t \rightarrow 0} \frac{m(t+\Delta t) - m(t)}{\Delta t} = -\frac{dm}{dt}$

we have

$m \cdot \frac{dv}{dt} + u \cdot \frac{dm}{dt} = 0$ ,

the ideal rocket’s flight equation obtained before in “Viva Rocketry! Part 1“.