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How long will it take Micrometeorites to cover the moon ?

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data
    If one micrometeorite (a sphere with a diameter of 1.50×10-6 m) strikes each square meter of the surface of the Moon each second (1 micrometeorite/m2/s), how many years will it take to cover the Moon to a depth of 1.80 m?
    Assume that after impact, the micrometeorites spread evenly over the surface of the moon.


    2. Relevant equations
    V = 4/3πr^3




    3. The attempt at a solution
    I'm not sure if I'm thinking through this correctly, but here it goes.

    SA of Moon (SA) = 3.8x10^13 m^2 (got this out of my textbook)
    Thickness (T) = 1.8 m (Given in problem statement)
    Total Volume needed to be filled by Micrometeorites = SA * T
    = (3.8x10^13)(1.8) = 6.84x10^13 m^3

    Volume of 1 Micrometeorite = V = 4/3πr^3 where r = (1.5x10^-6 / 2)
    = 4/3π(1.5x10^-6 / 2)^3 = 1.77x10^18

    Micrometeorites needed = Volume needed/Volume of Micrometeorites
    = (6.84x10^13)/(1.767x10^-18) = 3.87x10^31

    Since the rate at which they fall is 1 Micrometeorite/s that means a total of
    3.87x10^31 seconds

    which comes out to 1.22 years.


    Apparently this is wrong (we have to enter our answers online, and we're told if we're right or not). Please help!
     
  2. jcsd
  3. Sep 17, 2012 #2
    I'm not sure how you went from 3.87 x1031 seconds to 1.22 years

    should be ~1.23x1024 years
     
  4. Sep 17, 2012 #3
    You do not compute the volume of the spherical shell correctly.
     
  5. Sep 17, 2012 #4
    True about the years - I really messed that up. How am I supposed to calculate the spherical shell?
     
  6. Sep 17, 2012 #5
    The volume of the shell is the difference of two volumes.
     
  7. Sep 17, 2012 #6
    I keep trying but it's not working out. Could you give me a little more?
     
  8. Sep 17, 2012 #7
    well if the volume of the moon right now is 4/3πr3 and the volume of the moon after the micrometeorites add 1.8m to its radius is 4/3π(r + 1.8m)3

    then the difference between 4/3π(r + 1.8m)3 and 4/3πr3 will be the volume added by the micrometeorites

    so Vshell = 4/3π((r +1.8m)3 - r3)

    and then go from there
     
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