IB Physics HL D.1 Gravitational fields questions

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Question 1

Multiple choiceMedium1 mark

A satellite is at a distance r=2.0×107 mr=2.0\times10^7\text{ m} from the centre of a planet. The graph shows the planet’s gravitational field strength gg at selected distances rr What is the escape speed from this position?

0123405101520r/(107 m)g/(N kg-1)
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Answer: B. At r=2.0×107 mr=2.0\times10^7\text{ m} the graph gives g=5.0 N kg−1g=5.0\text{ N kg}^{-1} Since g=GM/r2g=GM/r^2 vesc=2GM/r=2gr=2(5.0)(2.0×107)=1.4×104 m s−1v_{\rm esc}=\sqrt{2GM/r}=\sqrt{2gr}=\sqrt{2(5.0)(2.0\times10^7)}=1.4\times10^4\text{ m s}^{-1}

A: uses the circular orbital speed gr\sqrt{gr} omitting the factor 2 under the square root.
C: assumes escape speed is twice the circular orbital speed, rather than 2\sqrt{2} times it.
D: reads g=20 N kg−1g=20\text{ N kg}^{-1} from the graph at r=1.0×107 mr=1.0\times10^7\text{ m} not the value at the satellite’s radius.

Remember: escape speed is 2\sqrt{2} times circular orbital speed at the same radius.

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