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A satellite of mass $m$ is at a distance $a$ from $a$ star of mass $M$. The speed of satellite is $u$. Suppose the law of universal gravity is $F = - G\frac{{Mm}}{{{r^{2.1}}}}$ instead of $F = - G\frac{{Mm}}{{{r^2}}}$, find the speed of the statellite when it is at $a$ distance $b$ from the star.
$\sqrt {{u^2} + 2GM\left( {\frac{1}{{{b^{1.1}}}} - \frac{1}{{{a^{1.1}}}}} \right)} $
$\sqrt {{u^2} + GM\left( {\frac{1}{{{a^{1.1}}}} - \frac{1}{{{b^{1.1}}}}} \right)}$
$\sqrt {{u^2} + \frac{2}{{1.1}}GM\left( {\frac{1}{{{b^{1.1}}}} - \frac{1}{{{a^{1.1}}}}} \right)}$
$\sqrt {{u^2} + \frac{2}{{2.1}}GM\left( {\frac{1}{{{b^{1.1}}}} - \frac{1}{{{a^{1.1}}}}} \right)}$
Solution
$E_{i}=E_{f}$
$\frac{1}{2} m u^{2}-U_{i}=\frac{1}{2} m v^{2}-U_{f}$
$U=-\int_{\infty}^{r} \frac{G M m}{r^{2.1}} d r$