## Task Condition

Determine the work (in joules) performed when lifting a satellite from the Earth's surface to a height of $H$ km. The mass of the satellite is $m$ tons, the radius of the Earth $R_{3}=6380$ km. The acceleration due to gravity $g$ at the Earth's surface is taken to be $10 \mathrm{~m} / \mathrm{c}^{2}$.

$$
m=7.0 \text { tons, } H=250 \text { km. }
$$
## Solution

By definition, the elementary work $d A=F(x) d x$, where $F(r)=G \frac{m \cdot M}{r^{2}} ; G=6.67 \cdot 10^{-11} \mathrm{H}^{*} \mathrm{m}^{*} \mathrm{m} /(\mathrm{kg}^{*} \mathrm{kg})$

\[
\begin{aligned}
& F_{0}=G \frac{m \cdot M}{R^{2}}=m g \\
& F_{x}=G \frac{m \cdot M}{(R+x)^{2}} \text{ - force of attraction at height } x \\
& \frac{F_{x}}{F_{0}}=\frac{G \frac{m \cdot M}{R^{2}}}{G \frac{m \cdot M}{(R+x)^{2}}}=\frac{R^{2}}{(R+x)^{2}} \Rightarrow F_{x}=\frac{F_{0} R^{2}}{(R+x)^{2}}=\frac{m g R^{2}}{(R+x)^{2}}=\frac{m g}{\left(1+\frac{x}{R}\right)^{2}} \\
& F(r)=\frac{m g}{\left(1+\frac{r}{R}\right)^{2}} \\
& A=\int_{0}^{H} F(x) d x=\int_{0}^{H} \frac{m g}{\left(1+\frac{x}{R}\right)^{2}} d x=7000 \cdot 10 \int_{0}^{250000} \frac{d x}{\left(1+\frac{x}{6380000}\right)^{2}}=7000 \cdot 10 \cdot 6380000 \int_{0}^{\frac{1}{250000} d\left(\frac{x}{6380000}+1\right)}\left(\frac{x}{6380000}+1\right)^{2}= \\
& =-\left.7000 \cdot 10 \cdot 6380000 \cdot \frac{x}{\frac{x}{6380000}+1}\right|_{0} ^{250000}=-7000 \cdot 10 \cdot 6380000\left(\frac{1}{\frac{250000}{638000}+1}-\frac{0}{\frac{0}{6380000}+1}\right) \approx 1.68 \cdot 10^{10} \text{ (J) }
\end{aligned}
\]

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\�\�\�\�\� $\% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB}$ $\%$ D1\%8B_22-12»

Categories: Kuznetsov Integral Problems 22 | Integrals | Problems for Checking

- Last edited: 09:41, 11 June 2009.
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## Problem Kuznetsov Integrals 22-13

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