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Homework Help: Find inequality for coefficient of restituition

  1. Jul 7, 2015 #1
    1. The problem statement, all variables and given/known data
    A small smooth sphere of mass 3 kg moving on a smooth horizontal plane with speed 8 ms-1 collides directly with a sphere of mass 12 kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e.

    2. Relevant equations
    v2 - v1/u1 - u2 =e

    3. The attempt at a solution
    To be honest, I'm totally lost. I know that v2-v1= 8e, and 0≤e≤1. Also, using principle of conservation of momentum, I know that v1 + 4v2=8. I don't know how to solve this because I only have two equations for 3 unknowns. Any hints?
     
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  3. Jul 7, 2015 #2

    haruspex

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    You have not used the constraint that after collision they move in opposite directions. What inequality does that give you?
     
  4. Jul 7, 2015 #3
    Is it v2 - v1 ≥ 8?
     
  5. Jul 7, 2015 #4

    haruspex

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    No. Which way will the heavier mass move?
     
  6. Jul 7, 2015 #5
    The heavier one will move to the right and take a positive velocity.
     
  7. Jul 7, 2015 #6

    haruspex

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    So what does that tell you about how the light mass moves off?
     
  8. Jul 7, 2015 #7
    The light mass moves in the opposite direction with a negative velocity
     
  9. Jul 7, 2015 #8

    haruspex

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    So write that as an inequality.
     
  10. Jul 7, 2015 #9
    I don't know....is the velocity of the lighter ball greater than velocity of heavier ball?
    So v1 - v2≥ 0?
     
  11. Jul 7, 2015 #10
    And less than 8?
     
  12. Jul 7, 2015 #11

    haruspex

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    Just write this statement
    as an inequality.
     
  13. Jul 7, 2015 #12
    Haha I'm feeling hopeless here. Please forgive my slowness. Is it -v1-v2≤8? I've tried this question for a long time...and I just can't get it.
     
  14. Jul 7, 2015 #13

    haruspex

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    The statement I've asked you to write as an inequality does not mention v2, so it will not feature in the inequality.
     
  15. Jul 7, 2015 #14
    Ok, I have made some headway, but in the right direction or not I'm not sure. So v1≤8. And if so, using the equation from conservation of momentum, I'll get v2≥4. Is it right?
     
  16. Jul 7, 2015 #15
    Ok, I have made some headway, but in the right direction or not. I'm not sure. So v1≤8. And if so, using the equation from conservation of momentum, I'll get v2≥4. Is it right?
     
  17. Jul 7, 2015 #16

    haruspex

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    If you are told x is negative, how would you write that as an inequality?
     
  18. Jul 7, 2015 #17
    So it's -v1 ≥ -8 which gives v1 ≥ 8?
     
  19. Jul 7, 2015 #18

    haruspex

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    Forget 8, forget about this whole problem for the moment. If you are told that some number x is negative, how do you express that statement as an inequality?
     
  20. Jul 7, 2015 #19
    n< 0
     
  21. Jul 7, 2015 #20

    haruspex

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    Right. So apply that to v1 and see where it leads.
     
  22. Jul 7, 2015 #21
    By substituting v1>0, into the equation from conservation of momentum, I get v2>2. Therefore, v2 -v1>2. Using this in coefficient of restitution equation yields e> 1/4
     
  23. Jul 7, 2015 #22

    haruspex

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    I assume you meant v1<0, but that answer looks right.
     
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