#### Question

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 × 10^{30} kg; mass of mars = 6.4 × 10^{23} kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 × 10^{8}kg; G= 6.67 × 10^{–11} m^{2}kg^{–2}.

#### Solution 1

Mass of the spaceship, *m*_{s }= 1000 kg

Mass of the Sun, *M* = 2 × 10^{30} kg

Mass of Mars, *m*_{m} = 6.4 × 10 ^{23} kg

Orbital radius of Mars, *R* = 2.28 × 10^{8 }kg =2.28 × 10^{11}m

Radius of Mars, *r *= 3395 km = 3.395 × 10^{6} m

Universal gravitational constant, G = 6.67 × 10^{–11} m^{2}kg^{–2}

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

= `(-GMm)/R`

Potential energy of the spaceship due to the gravitational attraction of Mars

= `(-GM_mm_s)/r`

Since the spaceship is stationed on Mars, its velocity and hence, its kinetic energy will be zero.

Total energy of the spaceship = `(-GMm)/r = (-GM_sm_m)/r`

`= -Gm_s(M/R+m_m/r)`

The negative sign indicates that the system is in bound state.

Energy required for launching the spaceship out of the solar system

= – (Total energy of the spaceship)

`=Gm_s(M/R + (m_m)/r)`

`=6.67xx10^(-11)xx 10^3 xx (2xx10^30)/(2.28xx10^(11))+ (6.4xx10^(23))/(3.395xx10^6)`

`=6.67xx10^(-8)(87.72xx10^(17)+1.88xx10^(17))`

`=6.67xx10^(-8)xx89.50xx10^(17)`

`=596.97xx10^9`

`=6xx10^(11)J`

#### Solution 2

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 gravitational attraction of the Sun = – GM m/R Potential energy of space-ship due to gravitational attraction of Mars = – G M’ m/R’ Since the K.E. of space ship is zero, therefore

total energy of spaceship =` -(GMm)/R - (GM'm)/R = -Gm(M/R + (M')/(R'))`

:. Energy required to rocket out the spaceship from the solar system = -(total energy of spaceship)

`=-[-Gm(M/R + (M')/R')] = Gm [M/R+(M')/R]`

`= 6.67xx10^(-11)xx1000xx [(2xx10^(30))/(2.28xx10^(11) )+(6.4xx10^(23))/(3395xx10^(3))]`

`= 6.67xx10^(-8) [20/2.28 + 6.4/33.95] xx 10^(18) J = 5.98 xx 10^(11) J`