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 spaceship = 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 × 10⁻¹¹ N m² kg⁻².

Answer 1

Given, Mass of the spaceship, $m_s$ = 1000 kg
Mass of the Sun, M = 2 × 10^30 kg
Mass of Mars, $m_m$ = 6.4 × 10²³ kg
Orbital radius of Mars, R = 2.28 × 108 kg =2.28 × 10¹¹m
Radius of Mars, r = 3395 km = 3.395 × 10^6 m


Potential energy of the spaceship due to the gravitational attraction of the Sun = -GM$m_s$/R

Potential energy of the spaceship due to the gravitational attraction of Mars = -G$m_mm_s$/r
Since the spaceship is stationary on Mars, its velocity = 0 and hence, its kinetic energy will be zero.

Total energy of the spaceship =  -GM$m_s$/R –  G$m_mm_s$/r
= -G$m_s$[ (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)

= G$m_s$[ (M / R) + ($m_m$ / r) ]

Now, putting, values of given all terms

= 6.67 × 10^-11 × 1000 × [ (2 × 10^30 / 2.28 × 10¹¹) + (6.4 × 10²³ / 3.395 × 10^6 ) ]

= 596.97 × 10^9  ≈  6 × 10¹¹ J.