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Question

What happened to the work done by friction BEFORE it touched the spring? Why was that neglected? Also they say there is no physical meaning behind the negative root, so what is the "unphysical" meaning behind the negative root? How are we supposed to know that the speed is constant before and just as it makes contact with the spring?

@atomSmasher

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I am referring to the green region. Shouldn't it be

$E_i = \frac{1}{2}mv_A ^2 - f_k(x_b + x_{green distance})$

$E_f = \frac{1}{2}kx_B ^2$

I realize we would have two unknowns then.

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The problem ignores the friction before it hits the spring because it is not relevant to the question. The question asks for the maximum compression of the spring. The force of friction prior to the spring is irrelevant because you are given the velocity at the point of impact.

The friction before impact would be relevant if you weren't given the velocity.

I am not sure what you mean about the physical meaning. The use of one or both of the quadratic roots depends on the setup of the origin and its axis. A negative number will not be part of a quadratic solution with these types of problems, usually! In this problem a negative value will not be considered because it doesn't make sense for the problem since we know the spring moved to the right (in the positive direction). Therefor -.25 cannot be used.

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I know it can't be used, but is there a meaning behind it? I am not asking for a physical meaning. We could physically see that there is no meaning behind the negative root, but the math is telling us that "hey you have two solutions!" –  sidht Dec 25 '11 at 20:33
    
Ah never mind, the -0.25 means if I had compressed the block and spring by -0.25 from x = 0 and releasing it to get v = 1.2m/s (at x = 0). I just tested with some numbers. Is there a way to intuitively know this? –  sidht Dec 25 '11 at 20:40
    
I see what you mean. The math is telling us we have two solutions because it just so happens to be in the form of a quadratic equation. Obviously that means there is symmetry about the y axis or y line. The important part to consider is the domain of the problem. The quadratic roots show two equations for a negative domain. Unfortunately the equation doesn't know where you want to stop! You are only concerned about 0 -> infinity. Therefor there is really only one true solution to the problem. –  atomSmasher Dec 26 '11 at 2:32
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