Question

A package of mass 9 kg sits at the equator of an airless asteroid of mass 4.0 1020 kg and radius 5.7 105 m. we want to launch the package in such a way that it will never come back, and when it is very far from the asteroid it will be traveling with speed 168 m/s. we have a large and powerful spring whose stiffness is 2.1 105 n/m. how much must we compress the spring?

Answer 1

Grav. Potential at surface of the asteroid:

V = - G.Ma./ R 
V = (-) 6.67^-11 x 4.0^20kg / 5.7^5m .. .. V = (-) 4.681 *10 ^5 J/kg 

The GPE of the package on the asteroid = 9.0kg x (-) 4.681*10^5J/kg = (-) 4.21 ^5 J

This is the amount of energy required to come back the package to infinity.

The total energy that needs to be transported to the package:

GPE + KE(for 187m/s) 
Total energy required E = 4.21*10^5 + (½x 9.0kg x 168²) = 5.48 * 10^5 J 

When the required energy is to be complete by releasing a compressed spring,
Elastic PE stored in spring = ½.ke² = 5.48 * 10^5 J where e = compression distance

e = √ (2 x 5.48*10^5 / 2.1*10^5)

e = 2.28 m