Friday, 29 March 2019

newtonian mechanics - Conservation of Linear Momentum at the point of collision


This is a pretty basic conceptual question about the conservation of linear momentum.


Consider an isolated system of 2 fixed-mass particles of masses m1 and m2 moving toward each other with velocities v1(t) and v2(t) respectively.



Now conservation of momentum says that at any point during the particles' motion the quantity m1v1(t)+m2v2(t)=constant


With non-zero velocities and non-zero masses this constant will be non-zero.


Let us say the particles collide at time t0. At the point of collision, both particles have velocity zero. which would mean that the constant above will be zero. Contradiction.


I realize I might be going wrong in my reasoning at the point of collision.


In fact, I feel defining velocity at that point would not even make sense, since if one considers the displacement functions xi(t) i=1,2 of the particles, then t0 would represent a point of non-differentiability of xi(t) for i=1,2.


So assuming there are no collisions, by following the text-book derivation I can see why


m1v1(t)+m2v2(t)=C1

before the collision and m1v1(t)+m2v2(t)=C2
after the collision


would hold true, but not why C1=C2


Can someone help me in clearing this up?




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