A package of mass 7 kg sits at the equator of an airless asteroid of mass 6.0 10

Question

A package of mass 7 kg sits at the equator of an airless asteroid of mass 6.0 1020 kg and radius 4.1 105 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 151 m/s. We have a large and powerful spring whose stiffness is 3.0 105 N/m. How much must we compress the spring?

The base of an decorative rectangular feature with a volume of V 105 ft3 is to be constructed in a restaurant. The bottom is made of marble, the sides are made of glass, and the top is open. If marble costs five times as much (per unit area) as glass, find the dimensions of the feature that minimize the cost of the materials. (Assume that the length is greater than or equal to the width. Give your answers correct to at least three decimal places.) length ft width ft height ft

Explanation / Answer

G = gravitational constant = 6.674*10^-11
asteroid mass = M = 6.0*10^20 kg
asteroid radius = R = 4.1*10^5 m
k = spring constant = 3*10^5 N/m
conservation of energy: (neglecting any frictional loss of energy)
KEi + PEi = KEf + PEf
0 - GMm/R + 0.5*kx^2 = 0.5*mvf^2 + 0
kx^2 = mvf^2 + GMm/R

x = sqrt(mvf^2/k + 2*GMm/kR)

x = sqrt(7*151^2/(3*10^5) + 2*6.67*10^-11*6*10^20*7/(3*10^5*4.1*10^5))
x = 2.255 m

Ans: the spring needs to be compressed 2.255 m

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