s. You stand on a spherical asteroid of uniform density whose mass is 2 x 1016 k
ID: 1782847 • Letter: S
Question
s. You stand on a spherical asteroid of uniform density whose mass is 2 x 1016 kg and whose radius 10 km (1 x 104 m). These are typical values for small asteroids. (a) How fast do you have to throw a rock straight up so that it never comes back to the asteroid and ends up traveling at a speed of 5 m/s when it is very far away? Explain briefly. (b) Suppose you use a spring to launch a payload horizontally from the asteroid so that the payload ends up far from the asteroid, traveling at a speed of 5 m/s. The payload has a mass of 24 kg. If the spring is to be compressed initially an amount 1.3 m, what stiffness k, must the spring be designed to have?Explanation / Answer
Given,
m = 2 x 10^16 kg ; r = 10^4 m
a)vf = 5 m/s
from conservation of energy
KEi + Ui = KEf + Uf
1/2 m v^i2 + - G Mm/ri = 1/2 m v^2
vi^2 = vf^2 + 2 GM/ri
vi^2 = 5^2 + 2 x 6.67 x 10^-11 x 2 x 10^16/(10^4) = 291.8
vi = sqrt (291.8) = 17.08 = 17.1 m/s
HEnce, vi = 17.1 m/s
b)m = 24 kg ; x = 1.3 m
from conservation of energy
KEi + Ui = KEf + Uf
1/2 k x^2 - GMm/ri = 1/2 m vf^2
k = m/x^2 x (vf^2 + 2 GM/ri)
k = 24/1.3^2 [5^2 + 2 x 6.67 x 10^-11 x 2 x 10^16/(10^4) ] = 4160.9 N/m
Hence, k = 4160.9 N/m
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