What is undoubtedly making this problem hard for me is the fact that there are t

Question

What is undoubtedly making this problem hard for me is the fact that there are three unknown velocities and not just two which I am used to. I am not really sure how to manipulate the conservation of momentum equation (m1v1i + m2v2i = m1v1f + m2v2f) to solve for v1i, v1f, and v2f when knowing that v2i is equal to zero. I also know that part of the process of solving this problem has to do with the conservation of energy (v1i -v2i)= -(v1f -v2f), but I am just stumped of what to do since there are three unknowns. Thank you.

Explanation / Answer

Let m be thew mass of neutron.

Initial Kinetic energy of Neutron K1i = (1/2)m(v1i)^2

Initial KInetic energy of carbon K2i = (1/2)(12m)(0) = 0

Total Kinetic energy of the system Ki = 0 + (1/2)m(v1i)^2 = (1/2)m(v1i)^2    ------------(1)

Since it is a eleastic collision we can write

               Relative velocity of approach before collision = Relative velocitry of seperation after collision

                                                                       v1i - 0 = v2f - v1f

                                                                    v2f - v1f = v1i       

                                                                             v2f = v1i + v1f

Final Kinetic energy of the Carbon K2f = (1/2)(12m)(v2f)^2

                                                         = (1/2)(12m)(v1i + v1f)^2

Fraction of Kinetic energy transfered = [(1/2)(12m)(v1i + v1f)^2]/[(1/2)m(v1i)^2  ]

                                                       = 12 [1 + (v1f/v1i)^2]

Now if you substitute the values for v1f & v1i then you get the result.

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