Mathematical Background - Energy and Momentum Conservation Principles
The use of conservation laws in elastic collision theory is a useful tool for solving elastic collision 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 they will have velocities V1 and V2. The use of bold letters denotes that the velocities are vectors.
According to the conservation of momentum principle, the momentum pBEFORE before the collision is equal to the momentum pAFTER after the collision:
pBEFORE = pAFTER =>
p 1,BEFORE + p 2,BEFORE = p 1,AFTER + p 2,AFTER =>
m1u1 + m2u2 = m1V1 + m2V2(1)
For more information on the conservation of momentum principle, check out the article, Understanding the Conservation of the Momentum Principle.
According to the conservation of kinetic energy, the kinetic energy KBEFORE before the collision is equal to the kinetic energy KAFTER after the collision:
KBEFORE = KAFTER =>
K 1,BEFORE + K 2,BEFORE = K 1,AFTER + K 2,AFTER =>
(1/2)m1u12 +(1/2)m2u22 = (1/2)m1V12 +(1/2)m2V22(2)