3. Why is linear momentum conserved?

A particle of mass “m” moving at velocity “v” has a linear momentum of . Two basic particles having the same speed v approaching each other for a “head-on” collision (i.e., where the velocities are parallel to the line determined by the particle centers) have a momentum of  since the momenta are equal and opposite. After impact the velocities are reversed so that the sum of their momentum is . Thus, linear momentum is conserved during this collision.

If two particles approaching a head-on collision with unequal velocities and , say in the same direction, have a momentum . If these particles are viewed from a frame moving at velocity  in the same direction as and then particle 1 will be seen to have a velocity while particle 2 will be seen to have a velocity opposite that of particle 1 . Thus, in this frame of reference the particles are approaching with equal and opposite velocities.  During collision the velocities are simply reversed, particle 1 has velocity and in this moving frame particle 2 has velocity after the collision.  If we now view these two particles after collision in the initial frame we add the velocity  to each particle.  Thus particle 1 has velocity and particle 2 has velocity . We see that during a head-on collision that the velocities are simply interchanged. Thus sum of the linear momentum of the two particles is unchanged during collision. (Before collision the momentum was and after collision it is ) .

In the general case of linear momentum we select a rectangular coordinate system with the x-axis passing through the line of centers when the particles are in contact. The x-components of the particles’ velocities again are simply interchanged and the y- and z-components are not changed during collision. Thus we have the momentum equation before and after collision as:

where each term on one side of the equation has an exact same term on the right side, and vice versa.

            Since the linear momentum of particles do not change if they don’t collide and it does not change if they do collide, linear momentum is always the same for any system of isolated particles.