A team of astronauts is on a mission to land on and explore a large asteroid. In addition to collecting samples and performing experiments, one of their tasks is to demonstrate the concept of the escape speed by throwing rocks straight up at various initial speeds. With what minimum initial speed 𝑣esc will the rocks need to be thrown in order for them never to "fall" back to the asteroid? Assume that the asteroid is approximately spherical, with an average density 𝜌=4.10×106 g/m3 and volume 𝑉=2.63×1012 m3 . Recall that the universal gravitational constant is 𝐺=6.67×10−11 N·m2/kg2 .

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GivenDersity of Asterciid (\rho)=4.10 \times 10^{6} \mathrm{~g} / \mathrm{m}^{3}(P)=4.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}Volume of Asteroid (V)=2.63 \times 10^{12} \mathrm{~m}^{3}Universal Gravitational Constant (G)=6.67 \times 10^{-11} \mathrm{~N}-\mathrm{m}^{2} / \mathrm{kg}We know that\begin{array}{c}V=\frac{4}{3} \pi R^{3} \\2.63 \times 10^{12}=\frac{4}{3} \pi R^{3} \\R^{3}=\frac{3 \times 2.63 \times 10^{12}}{4 \times 3.14} \\R=8.56 \times 10^{3} \mathrm{~m}\end{array}Relation Between G and g is Givers by\begin{array}{c}g=\frac{G M}{R^{2}}=\frac{G P V}{R^{2}} \\g=\frac{6.67 \times 10^{-11} \times 4.10 \times 10^{3} \times 2.63 \times 10^{12}}{4 \times 3.14} \\g=9.8 \times 10^{-3} \mathrm{~m} / \mathrm{sec}^{2}\end{array}We know Escope velocity\begin{array}{l}V_{\text {esc }}=\sqrt{2 g R} \\V_{\text {esc }}=\sqrt{2 \times 9.8 \times 10^{-3} \times 8.56 \times 10^{3}} \\V_{e s c}=12.91 \mathrm{~m} / \mathrm{sec}\end{array} ...