A body of mass m_1 with initial velocity collides with a second body, of mass m_

Question

A body of mass m_1 with initial velocity collides with a second body, of mass m_2, initially at rest-i.e., in the LAB frame. The initial kinetic energy K_1 i of the first body must be "paid for," i.e., must be provided by some energy source. But it is the total initial kinetic energy of the two bodies in the center-of-mass (CM) frame which represents the energy available for reactions, to create new particles, et cetera. Calculate the ratio of this total energy to the initial LAB energy in terms of the masses m_1 and m_2.

Explanation / Answer

assume it is inelasitic collision

m1v = (m1+m2)v'

v' = m1v/(m1+m2)

KEi = m1v^2/2

KEf = (m1+m2)*v'^2/2

KEf = (m1+m2)*[m1^2v^2/(m1+m2)^2 ] /2

KEf = m1^2v^2/2(m1+m2)

KEf/KEi = 1/(m1+m2)

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