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If the mass of a planet is eight times the mass of the earth and its radius is twice the radius of the earth, what will be the escape velocity of that planet?

Last updated date: 13th Jun 2024
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Hint: escape velocity of earth 11.2$km/\sec $. The escape velocity depends only on the mass and size of the object from which something is trying to escape. The escape velocity from the Earth is the same for a pebble as it would be for the Space Shuttle.

Complete step by step answer:
 let the mass of earth be ${M_{earth}}$ and the mass of planet be${M_{planet}}$.
According to the question we know that ${M_{planet}} = 8{M_{earth}}$
Let radius of earth be ${r_{earth}}$and radius of the planet be ${r_{planet}}$
According to the question we know that ${r_{planet}} = 2{r_{earth}}$
Equation to calculate the escape velocity of a planet. ${v_{escape}} = \sqrt {\dfrac{{2G{M_{PLANET}}}}{{{R_{planet}}}}} $
Where G is gravitational G = Newton's universal constant of gravity$6.67 \times {10^{ - 11}}N/{m^2}/k{g^3}$
M= mass of the planet; R = radius of the planet.
So to find the escape velocity for our planet.
${v_{escape}} = \sqrt {\dfrac{{2G8{M_{earth}}}}{{2{R_{earth}}}}} $
${v_{escape}} = (\sqrt {\dfrac{8}{2}} ) \times \sqrt {\dfrac{{2G{M_{earth}}}}{{{R_{earth}}}}} $
We already know that the escape velocity of earth is 11.2km/sec.
${v_{escape}} = \sqrt {\dfrac{8}{2}} \times 11.2$
$\therefore {v_{escape}}= 22.4km/\sec $

The escape velocity of our new planet will be 22.4 km/sec.

Note: Other method: We can solve this question by equating the ratio of planet earth and our new planet.
So to find we know
${v_{escape}} = \sqrt {2gR} $
If mass of the planet is 8 times that of earth
$\dfrac{{{v_{planet}}}}{{{v_{earth}}}} = \sqrt {\dfrac{{{M_{planet}}}}{{{M_{earth}}}} \times \Rightarrow\dfrac{{{\operatorname{R} _{earth}}}}{{{R_{planet}}}}} $
\Rightarrow\dfrac{{{v_{planet}}}}{{{v_{earth}}}} = \sqrt {\dfrac{{8{M_{earth}}}}{{{M_{earth}}}} \times \Rightarrow\dfrac{{{R_{earth}}}}{{2{R_{earth}}}}} \\
$\Rightarrow\dfrac{{{v_{planet}}}}{{{v_{earth}}}} = 2$
$\Rightarrow{v_{planet}} = 2{v_{earth}}$
$\Rightarrow {v_{planet}} = 2 \times 11.2km/\sec $
\therefore{v_{planet}} = 22.4km/\sec \\