Use the classic formulas for the velocity of spheres after an elastic impact:
v1′=m1+m2(m1−m2)v1+2m2v2 and v2′=m1+m2(m2−m1)v2+2m1v1
WORK, POWER, ENERGY 139
The velocity of the spheres' centre of gravity after impact, from the formula obtained in the previous problem, will be expressed thus:
m1+m2m1v1′+m2v2′
Substitute for v1′ and v2′ in this expression and we shall easily obtain that the velocity of the sphere's centre of gravity after an elastic impact will equal
m1+m2m1v1+m2v2
which is the same as the velocity of the spheres' centre of gravity before impact.
If the impact is inelastic, both spheres move together, i.e. with the same velocity. Therefore their centre of gravity will move with this velocity, which is also expressed by the proportion
m1+m2m1v1+m2v2
Therefore the same rule holds good for an inelastic impact also.
Thus the momentum of the centre of gravity of a system of spheres is the same before and after impact. In particular, if the centre of gravity of the system is at rest before impact, it will also be stationary after impact (whether the impact is elastic or inelastic). What obtains for spheres, obtains also for any system of particles of matter. The law deduced by us here should be considered an alternative formulation of the law of conservation of momentum, i.e. if outside forces do not affect it, the centre of gravity of any system of particles of matter conserves its momentum and therefore also its velocity. This fact sometimes allows some mechanics problems to be solved more simply. We shall demonstrate this later.