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Proving magnitude of impulse on either spheres

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  1. Jul 7, 2015 #1
    1. The problem statement, all variables and given/known data
    A sphere of mass m is moving with a speed V along a horizontal straight line. It collides with an identical sphere of mass m moving along the same straight line with speed u (u<V). Show that the magnitude of impulse on either sphere is
    ½m(1+e)(V-u), where e is the coefficient of restitution between the two spheres.

    2. Relevant equations
    m1u1 +m2u2 = m1v1 +m2v2
    V2-V1/ u1-u2= e
    3. The attempt at a solution
    I calculated impulse on sphere with velocity V. I =m(V1-V). Using coefficient of restitution, I get I = m(u1-e(V-u)-V). Using equation formed from conservation of momentum, I get I =m[u-V1- e(V-u)]. I have trouble getting rid of the u and V1 terms. How do I go about this?
     
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  3. Jul 7, 2015 #2

    mfb

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    Your notation is confusing. How are V1 and so on related to the V and u given in the problem statement?
     
  4. Jul 7, 2015 #3
    V1 and u1 are the velocities after collision.
     
  5. Jul 7, 2015 #4

    mfb

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    What are V2 and u2 then?
     
  6. Jul 7, 2015 #5
    Oh, that's just the general formula.
     
  7. Jul 7, 2015 #6

    haruspex

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    I think you mean those are the velocities of the first ball, after and before collision respectively. In the context of the question, u2 = u, and u1 = V.
    Your two 'relevant equations' have two unknowns, the two velocities after impact. Use them to express those velocities in terms of the given velocities and e.
     
  8. Jul 8, 2015 #7
    Okay, to be clear, I define V and u as the initial speeds of the balls, and V1 and u1 as the speeds after collision.
    So, since impulse is the same for both balls but in opposite direction, I form an equation m(V1- V) = -m(u1-u). Therefore I get m(V1 + u1) =m(V =u). However, coefficient of restitution gives me u1-v1 = e(V-u), and the u1-V1 term is not the same as in the first equation, so I can't make a complete substitution. How do I go from here?
     
  9. Jul 8, 2015 #8

    mfb

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    You can solve the last equation for u1 and plug it into the first one. That allows to solve for V1 which is the last unknown in the system. Then you can calculate the magnitude of impulse.
     
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