Consider a classical inelastic collision of the form A + B rightarrow C + D. (Fo

Question

Consider a classical inelastic collision of the form A + B rightarrow C + D. (For example, this could be a collision such as Na + Cl rightarrow Na^+ + Cl^- in which two neutral atoms exchange an electron and become oppositely charged ions.) Show that the law of conservation of classical momentum is invariant under the Galilean transformation if and only if total mass is conserved - as is certainly true in classical mechanics. (We shall find in relativity that the classical definition of momentum has to be modified and that total mass is not conserved.)

Explanation / Answer

15.2

In case of Galilean transformation, x' = x-ut, x = x'+ut

where, the unprimed parameters are for an inertial frame S where the observer is, and primed parameters are for another frame S' which is moving with a velocity u with respect to the frame S.

for frame S, the momentum p =mv = m dx/dt = m d/dt(x'+ut) = mdx'/dt + mu = mv'+mu= p'+mu

v is the velocity of the particle in frame S and v' is that for S'.

Therefore, p' = m (v-u)

Again, v' = dx'/dt = d/dt (x-ut)= dx/dt -u = v-u

Therefore, p'= mv'

Therefore, p=mv, p'=mv' .

Thus we can see, if total mass m is conserved , then momentum p is invariant under galilean transformation.

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