Question

A spaceship is stationed on Mars. How much velocity must be given to 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.

Answer 1

Dear Student,

Mass of the spaceship, m_s = 1000 kg
Mass of the Sun,M = 2 × 10³⁰ kg
Mass of Mars, m_m = 6.4 × 10 ²³ kg
Orbital radius of Mars, R = 2.28 × 10⁸ kg =2.28 × 10¹¹m
Radius of Mars, r = 3395 km = 3.395 × 10⁶ m
Universal gravitational constant, G = 6.67 × 10^–11 m²/kg²
Potential energy of the spaceship due to the gravitational attraction of the Sun = -GMm_s​ /R
Potential energy of the spaceship due to the gravitational attraction of Mars = -GMm_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_s /R - GM_mm_s /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) ]
= 6.67 × 10^-11 × 10³ × [ (2 × 10³⁰ / 2.28 × 10¹¹) + (6.4 × 10²³ / 3.395 × 10⁶ ) ]
= 596.97 × 10⁹ = 6 × 10¹¹ J
Assume we give a v velocity given to the the spaceship-
So,
1/2 m_sv² = 6 × 10¹¹
v = 3.4 ​× 10⁴m/s


Regards
Manoj Singh