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Solution - 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 - CBSE (Science) Class 11 - Physics

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 × 1030 kg; mass of mars = 6.4 × 1023 kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 × 108kg; G= 6.67 × 10–11 m2kg–2.

Solution 1

Mass of the spaceship, m= 1000 kg

Mass of the Sun, M = 2 × 1030 kg

Mass of Mars, mm = 6.4 × 10 23 kg

Orbital radius of Mars, R = 2.28 × 10kg =2.28 × 1011m

Radius of Mars, = 3395 km = 3.395 × 106 m

Universal gravitational constant, G = 6.67 × 10–11 m2kg–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`

Is there an error in this question or solution?

APPEARS IN

 NCERT Physics Textbook for Class 11 Part 1 (with solutions)
Chapter 8: Gravitation
Q: 24 | Page no. 202

Reference Material

Solution for 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 × 1030 Kg; Mass of Mars = 6.4 × 1023 Kg; Radius of Mars = 3395 Km concept: Gravitational Potential Energy. For the course CBSE (Science)
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