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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

Verified by Experts

K.E. of spaceship `= (1)/(2) mv^(2)`
P.E. of spaceship in orbit `= (-GMm)/(R+h)`
Total energy of spaceship at height h
= K.E. + P.E.
`= (1)/(2) mv^(2) - (GMm)/(R + h)`
`= (1)/(2)m.(GM)/((R+h)) - (GMm)/(R+h)`
`E = - (GMm)/(2(R+h))`
m = mass of spaceship = 1,000 kg
`R = 2.28 xx 10^(8) xx 10^(3) m`
`h = 3,395 xx 10^(3)m`
M = mass of sun `= 2 xx 10^(30) kg`
`E = (6.67 xx 10^(-11) xx 2 xx 10^(30) xx 1,000)/(2(2.28 xx 10^(11) + 3,395 xx 10^(3)))`
`E = 2.9 xx 10^(11) J`
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