Gravitation

Question : 24

Total: 25

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the space ship = 1000 kg; mass of the Sun =2×1030 kg; mass of Mars =6.4×1023 kg; radius of Mars =3395 km; radius of the orbit of Mars =2.28×108 km; G=6.67×10–11 N m‌2 kg‌–2.

Solution:

Potential energy of spaceship due to gravitational attraction of the Sun =

−GMSm

Here, MS= Mass of Sun,
m = Mass of spaceship, R = radius of orbit of mars
Total potential energy of spaceship,
U=

−GMSm

−

GMmm

=−Gm[

]
Here, Mm= mass of Mars
Rm= radius of mars =[−Gm[

]]

=−6.67×10−11×1000×[

2×1030

2.28×1011

6.4×1023

3395×103

]

=−6.67×10−8[

2.28

6.4

33.95

]×1018J=−5.98×1011J

Let v be the orbital velocity of Mars,

mv2

r02

or v2=

Since, velocity of spaceship is same as that of Mars, therefore, K.E. of spaceship =

(m)v2=

(m)

GM(m)

2r0

or,

K.E.=

(6.67×10−11)×(2×1030)×1000J

2‌left(2.28×1011)

=2.925×1011J

Total energy of the spaceship, i.e.,

E=K.E.+U=2.925×1011J−5.98×1011J=−3.1×1011J

Negative energy denotes that the spaceship is bound to the solar system.Thus, energy needed by the spaceship to escape from the solar system =3.1×1011J

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