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A space-ship is stationed on Mars. How m...

A space-ship is stationed on Mars. How much energy must be expended on the spaceship to rocket it out of the solar system ? Mass of the spaceship `= 1000 kg`, Mass of the sun `= 2 xx 10^(30) kg`. Mass of the Mars `= 6.4 xx 10^(23) kg`, Radius of Mars `= 3395 km`. Radius of the orbit Mars `= 2.28 xx 10^(11) m, G = 6.67 xx 10^(-11) N m^(2) kg^(-2)`.

Text Solution

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Let `R`, be the radius of orbit of Mars and `R'` be the radius of the Mars. `M` be the mass of the sun and `M`' be the mass of mars. If `m` is the mass of the space-ship, then
Potential energy of space-ship due to gravitationl attraction of the sun `= - GM m//R`
Potential energy of space-ship due to gravitational attraction of Mars `= - GM' m//R`
since, the K.E. of space ship is zero, therefore,
total energy of space ship` = - (GM m)/(R ) - (GM' m)/(R') = - GM m((M)/(R ) + (M')/(R'))`
`:.` Energy required to rocket out the space ship from the solar system = - (total energy of space ship)
`= - [-Gm((M)/(R) + (M')/(R'))] = Gm[(M)/(R) + (M')/(R')]`
`= 6.67 xx 10^(-11) xx 1000 xx [(2xx 10^(30))/(2.28 xx 10^(11)) + (6.4 xx 10^(23))/(3395 xx 10^(3))]`
`= 6.67 xx 10^(-8) [(20)/(2.28) + (6.4)/(33.95)] xx 10^(18) J = 5.98 xx 10^(11) J`
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