A package of mass 5 kg sits at the equator of an airless asteroid of mass 3.0 1020 kg and radius 4.3 105 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 163 m/s. We have a large and powerful spring whose stiffness is 1.8 105 N/m. How much must we compress the spring?

 The Spring must be compressed by a length  

Explanation:

From the question we are told that

         The mass is

        The mass of the  asteroid

         The radius of the asteroid is

          The speed of the asteroid is  

         The stiffness is

From this question we can see that it the spring that provide the required energy to lunch this package

    Hence according to the law of conservation of energy

Energy Provided by the spring =  kinetic energy of the package +  the gravitational potential energy of the package

  Energy Provided by the spring  

Where x is the length the spring is to be compressed to obtain the desired result

  kinetic energy of the package  

the gravitational potential energy of the package  

Expressing the equation mathematically

Making x the subject  

Substituting values