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. Question 8.24 Graviotation

Answer 1

Mass of the spaceship, ms =1000 kg
Mass of the Sun, M = 2 ×10^30 kg
Mass of Mars, Mm = 6.4 x 10^23 kg
Orbital radius of Mars, R- 2.28 108 kg -2.28 x 10 ^11 m
Radius of Mars, r = 3395 km = 3.395 x 10^6m
Universal gravitational constant, G=6.67 10^- I1 m²kg-²

Potential Energy of the spaceship due to

Gravitational attraction of the sun = -GMms/R

Potential energy of spaceship due to attraction of Mars = -GMm × ms/r

As the spaceship is stationary then it's velocity must be zero means no kinetic energy .
so : Total energy of the spaceship will be

= -GMms/R - (-GMm ms/r)
= -Gms { M/R +Mm/r)
= Now putting the values

=> 6.67×10-¹¹×10³ (2×10^30/2.28×10¹¹ + 6.4×10²³/3.95×10^6)


Solving this you will get
= 6×10¹¹J

Answer 2

PE of spaceship due to sun = -GMm(s) / R 

PE of spaceship die to Mars = -GM(m) m(s)/r

KE = 0 

Total Enrgy = -GMm(s) + Gm(s) M(m) /r

= -GM(s)[ M/R + M(m) / r] 

Energy = Gm(s) [M/R + M(m) /r]

= 6.67 x 10^-11 x 10^3 [ 2 x 10^30 / 2.28 x 10^11 + 6.4 x 10^23 / 3.395 x 10^6]

= 6.67 x 10^-8 x 89.56 x 10^17

= 596.6 x 10^9 

= 6.0 x 10^11 J (aprox)