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Math Help - Mechanics, collisions help

  1. #1
    Super Member craig's Avatar
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    Mechanics, collisions help

    Hi, not been online in some time but back again now with some more probably simple questions

    I can't get my head around this question involving collisions, it's a bit wordy to please bear with me.

    3 smooth identical spheres each of mas 1kg move on a straight line on a smooth horizontal floor. B lies between A and C.

    Spheres A and B are projected towards each other with speeds 3 m/s and 2 m/s respectivly, and C is projected away from sphere A at a speed of 2m/s.

    The coefficient of restitution between any two spheres is e, show that sphere B will only collide with C if e > \frac{3}{5}
    Obviously the final velocity of B needs to be greater than 2, but I'm having trouble making this into an equation?

    Thanks in advance

    Craig
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  2. #2
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    Grandad's Avatar
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    Coefficient of Restitution

    Hello Craig

    There are two principles involved here. They are momentum and restitution. In the equations that follow, I am taking the direction A \rightarrow C as positive, and I'm assuming that the velocities of A and B are v_a and v_b respectively, after the impact. So:

    1 The momentum of the system is conserved at the impact. So (noting that the masses are all 1 kg):

    3 - 2 = v_a + v_b

    \Rightarrow v_a + v_b = 1 (1)

    2 The restitution equation is: the velocity of separation = -e times the velocity of approach.

    In other words, the (velocity of B - velocity of A) after impact = -e \times (velocity of B - velocity of A) before impact. So (remembering that we are taking A \rightarrow C as the positive direction):

    v_b - v_a = -e(-2 - 3) = 5e (2)

    Add (1) and (2):

    2v_b = 1 + 5e

    and v_b > 2 if B collides with C

    \Rightarrow 1 + 5e > 4

    \Rightarrow e > \frac{3}{5}

    Grandad
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  3. #3
    Super Member craig's Avatar
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    Thank you

    Really simple when you put it like that, I can solve these problems quite easily when there's numbers involved, but it's when the velocitys are Us and Vs that I have to start thinking.

    Thanks again

    Craig
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