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¯¹¹N m² kg₂.
Potential energy of the spaceship due to the gravitational attraction of the Sun is given by,
V s = − G M m R
Where, m is the mass of the spaceship, M is the mass of sun, and R is the orbital radius of mars.
Potential energy of the spaceship due to the gravitational attraction of Mars is given by,
V M = − G M m m r
Where, M m is the mass of mars and r is the radius of mars.
Total potential energy of the spaceship is given as,
V = V s + V M = − G M m R + ( − G M m m r )
Energy required for launching the spaceship out of the solar system is equal to the total energy of the spaceship.
E = G m ( M R + M m r )
By substituting the given values in the above expression, we get
E = 6.67 × 10 − 11 × 10 3 × ( 2 × 10 30 2.28 × 10 11 + 6.4 × 10 23 3.395 × 10 6 ) = 6.67 × 10 − 11 × 10 3 × ( 87.72 × 10 17 + 1.88 × 10 17 ) = 6.67 × 10 − 11 × 10 3 × 89.50 × 10 17 ≈ 6 × 10 11 J
Thus, the energy required for launching the spaceship out of the solar system is 6 × 10 11 J .
Potential energy of the spaceship due to the gravitational attraction of the Sun is given by,
Where,
Potential energy of the spaceship due to the gravitational attraction of Mars is given by,
Where,
Total potential energy of the spaceship is given as,
Energy required for launching the spaceship out of the solar system is equal to the total energy of the spaceship.
By substituting the given values in the above expression, we get
Thus, the energy required for launching the spaceship out of the solar system is
