Question 2: Gravitation

Question

Two satellites $$A$$ and $$B$$ revolve round the same planet in coplanar circular orbits lying in the same plane. Their periods of revolutions are $$1 \mathrm{~h}$$ and $$8 \mathrm{~h}$$, respectively. The radius of the orbit of $$A$$ is $$10^4 \mathrm{~km}$$. The speed of $$B$$ is relative to $$A$$. When they are closed in $$\mathrm{km} / \mathrm{h}$$ is

Accepted Answer

From Kepler's law,
$$\begin{array}{ll} 
& \frac{T_A^2}{T_B^2}=\frac{r_A^3}{r_B^3} \Rightarrow \frac{1^2}{8^2}=\frac{\left(10^4ight)^3}{r_B^3} \
\Rightarrow & r_B^3=64 	imes\left(10^4ight)^3 \
	herefore & r_B=4 	imes 10^4 \mathrm{~km}
\end{array}$$
Speed of satellite $$A$$,
$$v_A=\frac{2 \pi r_A}{T_A}=\frac{2 \pi 	imes 10^4}{1}$$
$$=2 \pi 	imes 10^4 \mathrm{~km} / \mathrm{h}$$
Speed of satellite B,
$$\begin{aligned}
& v_B=\frac{2 \pi r_B}{T_B}\
& =\frac{2 \pi 	imes 4 	imes 10^4}{8} \
& =\pi 	imes 10^4 \mathrm{~km} / \mathrm{h}
\end{aligned}$$
The speed of $$B$$ relative to $$A$$ when they are closed.
$$\begin{aligned}
v_{B A} & =v_A-v_B \
& =2 \pi 	imes 10^4-\pi 	imes 10^4 \
& =\pi 	imes 10^4 \mathrm{~km} / \mathrm{h}
\end{aligned}$$

Answer

$$3 \pi 	imes 10^4$$

Answer

zero

Answer

$$2 \pi 	imes 10^4$$

Answer

$$\pi 	imes 10^4$$

Gravitation

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