# Show that moon would depart forever if its speed were increased by 42%

${v}_{0}=\sqrt{\frac{GM}{R}}$

Expression for Escape velocity:

${v}_{e}=\sqrt{\frac{2GM}{r}}$

From, the above expressions, it is clear that ${v}_{e}=\sqrt{2}{v}_{o}$

Escape velocity: It is the velocity required by moon to escape or depart forever.

Percentage increase of velocity required to reach terminal velocity.

$Percentageoincrea\mathrm{sin}=\left(\frac{{v}_{e}-{v}_{0}}{{v}_{0}}\right)$(100)

$=\left(\frac{\sqrt{2}{v}_{0}-{v}_{o}}{{v}_{0}}\right)\left(100\right)\phantom{\rule{0ex}{0ex}}=\left(\frac{1.414{v}_{0}-{v}_{0}}{{v}_{0}}\right)100\phantom{\rule{0ex}{0ex}}=41.4\%\phantom{\rule{0ex}{0ex}}$

Hence, it is proved that moon would depart forever if its speed were raised by 42%.

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