Piece of Pi

A while back, we deemed that it is utterly impractical to calculate the value of $\pi$ using the partial sum of Leibniz’s series due to its slow convergence (see “Pumpkin Pi“)

Fig. 1

As illustrated in Fig. 1, in order to determine each additional correct digit of $\pi$ , the number of terms in the summation must increase by a factor of 10.

What we need is a fast converging series whose partial sum yields given number of correct digits with far fewer terms.

Looking back, we see that the origin of Leibniz’s series is the definite integral

$\frac{\pi}{4} = \int\limits_{0}^{1}\frac{1}{1+x^2} dx\quad\quad\quad(1)$

To find the needed new series, we consider a variation of (1), namely,

$\frac{\pi}{6} = \int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{1}{1+x^2} dx\quad\quad\quad(2)$

Given the fact (see “Pumpkin Pi“) that

$\frac{1}{1+x^2} = \sum\limits_{k=1}^{n}(-1)^{k+1}x^{2k-2}+\frac{(-1)^n x^{2n}}{1+x^2}\quad\quad\quad(3)$

We proceed to integrate (3) with respect to $x$ from $0$ to $\frac{1}{\sqrt{3}}$ ,

$\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{1}{1+x^2} dx=\int\limits_{0}^{\frac{1}{\sqrt{3}}}\sum\limits_{k=1}^{n}(-1)^{k+1}x^{2k-2} dx + \int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{(-1)^n x^{2n}}{1+x^2}dx$

$= \sum\limits_{k=1}^{n}(-1)^{k+1}\int\limits_{0}^{\frac{1}{\sqrt{3}}}x^{2k-2} dx + (-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx$

$= \sum\limits_{k=1}^{n}(-1)^{k+1}\frac{x^{2k-1}}{2k-1}\bigg|_{0}^{\frac{1}{\sqrt{3}}}+ (-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx$

As result, (2) becomes

$\frac{\pi}{6}=\sum\limits_{k=1}^{n}(-1)^{k+1}\frac{(\frac{1}{\sqrt{3}})^{2k-1}}{2k-1}+(-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx$

$= \sum\limits_{k=1}^{n}(-1)^{k+1}\frac{{(\frac{1}{3}})^k}{(2k-1)\frac{1}{\sqrt{3}}} + (-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx$

$= \sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)} + (-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx$

i.e.,

$\frac{\pi}{6} - \sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)} = (-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx\quad\quad\quad(4)$

By (4),

$|\frac{\pi}{6} - \sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)}| = |(-1)^n\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx|=\int\limits_{0}^{\frac{1}{\sqrt{3}}}\frac{x^{2n}}{1+x^2}dx<\int\limits_{0}^{\frac{1}{\sqrt{3}}}x^{2n}dx$

$=\frac{x^{2n+1}}{2n+1}\bigg|_0^{\frac{1}{\sqrt{3}}}$

$=\frac{1}{3^n \sqrt{3} (2n+1)}$

which gives

$|\sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)} - \frac{\pi}{6}| < \frac{1}{3^n \sqrt{3} (2n+1)}$ .

And so

$-\frac{1}{3^n \sqrt{3} (2n+1)}<\sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)}-\frac{\pi}{6}<\frac{1}{3^n \sqrt{3} (2n+1)}\quad\quad\quad(5)$

Since $\lim\limits_{n\rightarrow \infty}\frac{1}{3^n \sqrt{3} (2n+1)}=0$ , (5) implies

$\lim\limits_{n\rightarrow \infty} \sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)}-\frac{\pi}{6}= 0$ .

Hence,

$\lim\limits_{n\rightarrow \infty} \sqrt{3}\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{3^k(2k-1)}=\frac{\pi}{6}$ .

It follows that the value of $\pi$ can be approximated by the partial sum of a new series

$6\sqrt{3}\sum\limits_{k=1}^{\infty}\frac{(-1)^{k+1}}{3^k(2k-1)}$

Let’s compute it with Omega CAS Explorer (see Fig. 2, 3)

Fig. 2

Fig. 2 shows the series converges quickly. The sum of the first 10 terms yields the first 6 digits!

Fig. 3

Totaling the first 100 terms of the series gives the first 49 digits of $\pi$ (see Fig. 3)

Exercise 1. Show that $\lim\limits_{n\rightarrow \infty}\frac{1}{3^n \sqrt{3} (2n+1)}=0$ .

Exercise 2. Can we use $\frac{\pi}{3} = \int\limits_{0}^{\sqrt{3}}\frac{1}{1+x^2}dx$ to compute the value of $\pi$ in a similar fashion? Explain.

Viva Rocketry! Part 2

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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.

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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 (Viva Rocketry! Part 1.3)“) 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

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

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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 $\beta = \frac{1}{100}$ ,

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

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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. 7

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. 8

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.

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Fig. 9

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

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

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Fig. 10

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Fig. 11

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 (Viva Rocketry! Part 1.4)

An alternative derivation of ideal rocket’s flight equation (Viva Rocketry! Part 1.3)

2 Replies

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“.

Does gravity matter ? (Viva Rocketry! Part 1.2)

Thunderbolt (Viva Rocketry! Part 1.1)

Viva Rocketry! Part 1

4 Replies

In this post, we will first look at the main characteristics of rocket flight, and then examine the feasibility of launching a satellite as the payload of a rocket into an orbit above the earth.

A rocket accelerates itself by ejecting part of its mass with high velocity.

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Fig. 1

Fig. 1 shows a moving rocket. At time $t+\Delta t$ , the mass $\Delta m$ leaves the rocket in opposite direction. As a result, the rocket is being propelled away with an increased speed.

Let

$m(\square), m_{\square}$ – the mass of rocket at time ${\square}$

$\vec{v}_{\square}$ – the velocity of rocket at time $\square$

$v(\square), v_{\square}$ – the magnitude of $\vec{v}_{\square}$

$\vec{v}^*_{t+\Delta t}$ – the velocity of ejected mass $\Delta m$ at $t + \Delta t$

$v^*_{t+\Delta t}$ – the magnitude of $\vec{v}^*_{t+\Delta t}$

$u$ – the magnitude of $\Delta m$ ‘s velocity relative to the rocket when it is ejected. It is time invariant.

From Fig. 1, we have

$\Delta m = m_t - m_{t + \Delta t}$ ,

$\vec{v}_t = v_{t}$ ,

$\vec{v}_{t + \Delta t} = v_{t + \Delta t}$

and most notably, the relationship between $v^*_{t+\Delta t}, v_{t+\Delta t}$ and $u$ (see “A Thought Experiment on Velocities”):

$v^*_{t+\Delta t} = u - v_{t + \Delta t}$ .

It follows that

$\vec{v}^*_{t+\Delta t} = -v^*_{t+\Delta t} = v_{t + \Delta t} - u$ ,

momentum at time $t$ : $\vec{p}(t) = m_t \vec{v}_t = m_t v_t$

and,

momentum at time $t+\Delta t$ : $\vec{p}(t+\Delta t) = m_{t+\Delta t}\vec{v}_{t+\Delta t} + {\Delta m} \vec{v}^*_{t+\Delta t}=m_{t+\Delta t}\vec{v}_{t+\Delta t} + (m_t - m_{t+\Delta t}) \vec{v}^*_{t+\Delta t}$ $= m_{t+\Delta t}v_{t+\Delta t} + (m_t - m_{t+\Delta t})(v_{t+\Delta t}-u)$ .

Consequently, change of momentum in $\Delta t$ is $\vec{p}(t+\Delta t)- \vec{p}(t) = m_t (v_{t + \Delta t} - v_t) + u (m_{t + \Delta t} - m_t)$ .

Apply Newton’s second law of motion to the whole system,

$\vec{F}= {d \over dt} \vec{p}(t)$

$= \lim\limits_{\Delta t \rightarrow 0} {{\vec{p}(t+\Delta t) - \vec{p}(t)} \over \Delta t}$

$= \lim\limits_{\Delta t \rightarrow 0} { {m_t (v_{t + \Delta t} - v_t) + u (m_{t + \Delta t} - m_t)} \over {\Delta t} }$

$= \lim\limits_{\Delta t \rightarrow 0} {m_t {{v_{t + \Delta t} - v_t} \over {\Delta t}} + {u {{m_{t + \Delta t} - m_t} \over {\Delta t}}}}$

$= m_t \lim\limits_{\Delta t \rightarrow 0}{(v_{t+\Delta t} - v_t) \over {\Delta t}} + u \lim\limits_{\Delta t \rightarrow 0}{(m_{t +\Delta t} - m_t) \over \Delta t}$

That is,

$\vec{F} = m(t) {d \over dt} v(t) + u {d \over dt} m(t)$

where $\vec{F}$ is the sum of external forces acting on the system.

To get an overall picture of the rocket flight, we will neglect all external forces.

Without any external force, $\vec{F} = 0$ . Therefore

$0 = m(t) {d \over dt} v(t) + u {d \over dt} m(t)$

i.e.,

${d \over dt} v(t) = -{u \over m(t)} {d \over dt} m(t)\quad\quad\quad(1)$

That fact that $u, m(t)$ in (1) are positive quantities shows as the rocket loses mass ( ${d \over dt} m(t) < 0$ ), its velocity increases ( ${d \over dt} v(t) > 0$ )

Integrate (1) with respect to $t$ ,

$\int {d \over dt} v(t)\;dt = -u \int {1 \over m(t)} {d \over dt} m(t)\;dt$

gives

$v(t) = -u \log(m(t)) + c$

where $c$ is the constant of integration.

At $t = 0, v(0)=0, m(0) = m_1 + P$ where $m_1$ is the initial rocket mass (liquid or solid fuel + casing and instruments, exclude payload) and $P$ the payload.

It means $c = u \log(m_1+P)$ .

As a result,

$v(t) = -u \log(m(t)) + u \log(m_1+P)$

$= -u (\log(m(t) - \log(m_1+P))$

$= -u \log({m(t) \over m_1+P})$

i.e.,

$v(t) = -u \log({m(t) \over m_1+P})\quad\quad\quad(2)$

Since $m_1$ is divided into two parts, the initial fuel mass $\epsilon m_1 (0 < \epsilon < 1)$ , and the casing and instruments of mass $(1-\epsilon)m_1$ , $m(0)$ can be written as

$m(0) = \epsilon m_1 + ( 1 - \epsilon) m_1 + P$

When all the fuel has burnt out at $t_1$ ,

$m(t_1) = (1 - \epsilon)m_1 + P$

By (2), the rocket’s final speed at $t_1$

$v(t_1) = -u \log({m(t_1) \over {m_1+P}})$

$= -u \log({(1-\epsilon)m_1+P \over {m_1 + P}})$

$= -u \log({{m_1 + P -\epsilon m_1} \over {m_1+P}})$

$= -u \log(1-{{\epsilon m_1} \over {m_1+P}})$

$= -u \log(1-{\epsilon \over {1 + {P \over m_1}}})$

$= -u \log(1-{\epsilon \over {1 + \beta}})$

where $\beta = {P \over m_1}$ .

In other words,

$v(t_1) =-u \log(1-{\epsilon \over {1 + \beta}})\quad\quad\quad(3)$

Hence, the final speed depends on three parameters

$u, \epsilon$ and $\beta$

Typically,

$u = 3.0\;km\;s^{-1}, \epsilon = 0.8$ and $\beta = 1/100$ .

Using these values, (3) gives

Screen Shot 2018-11-05 at 9.21.14 PM.png

$v_1 = 4.7\;km\;s^{-1}\quad\quad\quad(4)$

This is an upper estimate to the typical final speed a single stage rocket can give to its payload. Neglected external forces such as gravity and air resistance would have reduced this speed.

With (4) in mind, let’s find out whether a satellite can be put into earth’s orbit as the payload of a single stage rocket.

We need to determine the speed that a satellite needs to have in order to stay in a circular orbit of height $h$ above the earth, as illustrated in Fig. 2.

Screen Shot 2018-11-08 at 3.47.09 PM.png

Fig. 2

By Newton’s inverse square law of attraction, The gravitational pull on satellite with mass $m_{s}$ is

${\gamma \; m_{s} M_{\oplus} \over (R_{\oplus} + h)^2}$

where universal gravitational constant $\gamma = 6.67 \times 10^{-11}$ , the earth’s mass $M_{\oplus} = 5.9722 \times 10^{24}\; kg$ , and the earth’s radius $R_{\oplus} = 6371\;km$ .