Hey could I please get some help on this problem? Thank you! A package of mass 1
ID: 1340198 • Letter: H
Question
Hey could I please get some help on this problem? Thank you!
A package of mass 12 kg sits on the surface of an airless asteroid of mass 1.7 times 10^14 kg and radius 2.0 times 10^3 m. The package is launched by a large, powerful, low mass spring, whose stiffness is 315 N/m. After launch, the package is observed to be moving at a speed of 3.0 m/s when it is 5 times 10^3 m from the center of the asteroid. How much was the spring compressed to launch the package? Start from a fundamental principle and clearly show every step in your reasoning.Explanation / Answer
Gravitational Potential at the surface of the asteroid, V = -GMasteroid / R
V = - (6.67*10-11*1.7*1014) / (2.0*103) = - 11339 / 2000 = - 5.6695 J/kg
The Gravitational Potential Energy on the asteroid, GPE = -Vmpackage = - 5.6695*12 J = - 68.0340 J
This is the amount of energy required to make the to be thrown by the spring, against the gravitational force of attraction of the asteroid.
The total energy that needs to be transferred to the package is GPE + (K.E. for 3.0 m/s)
Total Energy E = 68.0340+(0.5*12*32) = 122.0340 J
when the required energy is to be supplied by releasing a compressed spring, then it would be equal to the elastic potential energy stored in the spring = 1/2 * K e2 (where K is the stiffness, e is the compression)
Therefore, E = 1/2*Ke2
e = sqrt(2E/K) = sqrt(0.7748) = 0.8802 m