Mathematical Background - Energy and Momentum Conservation Principles
The use of conservation laws in inelastic collision theory is a useful tool for solving these problems. Lets assume that we have a system of two ideal particles with masses m1 and m2 moving in two dimensions. No external forces are acting on the system (closed) and the two masses have initial velocities u1 and u2, respectively. After the collision, their velocities will be V1 and V2, respectively (see image below). Bold letters denote vectors.
According to the conservation of momentum principle, the momentum pBEFORE before the collision is equal to the momentum pAFTER after the collision:
p BEFORE = p AFTER =>
p 1,BEFORE + p 2,BEFORE = p 1,AFTER + p 2,AFTER =>
m1u1 + m2u2 = m1V1 + m2V2
For more information on the conservation of momentum principle, read the article, Understanding the Conservation of the Momentum Principle.
The conservation of kinetic energy K does not apply in the case of inelastic collision. A part of this kinetic energy will be converted to heat or other forms of energy (ΔE). However, the conservation of energy is still valid provided the system is closed:
K BEFORE = K AFTER + ΔE =>
K 1,BEFORE + K 2,BEFORE = K 1,AFTER + K 2,AFTER + ΔΕ=>
(1/2)m1u12 + (1/2)m2u22 = (1/2)m1V12 + (1/2)m2V22 + ΔE
The last expression can be used to calculate the amount of lost energy ΔΕ, when all other parameters are known.
The diagram below shows the general case of an inelastic collision between two objects. In this particular diagram, the velocities of both objects have decreased. (Click to enlarge)