Apackage of mass 5 kg sits on an airless asteroid of mass 7.6 × 1020 kg and radius 8.0 × 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 170 m/s. we have a large and powerful spring whose stiffness is 2.8 × 105 n/m. how much must we compress the spring?

s =  1.7 m

Explanation:

from the question we are given the following:

Mass of package (m) = 5 kg

mass of the asteriod (M) = 7.6 x 10^{20} kg

radius = 8 x 10^5 m

velocity of package (v) = 170 m/s

spring constant (k) = 2.8 N/m

compression (s) = ?

Assuming that no non conservative force is acting on the system here, the initial and final energies of the system will be the same. Therefore  

•Ei = Ef

•Ei = energy in the spring + gravitational potential energy of the system

•Ei = \frac{1}{2}ks^{2} + \frac{GMm}{r}

•Ef = kinetic energy of the object

•Ef = \frac{1}{2}mv^{2}  

•\frac{1}{2}ks^{2} + (-\frac{GMm}{r}) = \frac{1}{2}mv^{2}  

•s =

s =

s =  1.7 m