# Notes

Doasap
Conservation of Energy and Linear Momentum in Collisions

← Total kinetic energy before the collision and the total kinetic energy after the collision is the same or energy is conserved ( elastic collision

½ m1 v1i 2   +   ½ m2 v2i 2   =   ½ m1 v1f 2   + ½ m2 v2f 2           …………[1]

← Collisions in which kinetic energy is not conserved are said to be inelastic collisions.

← From conservation of momentum
m1 v1i   +   m2 v2i   =   m1 v1f   +   m2 v2f               …………[2]
2 equations, solve for two unknowns.
Rewrite the momentum equations as
m1 ( v1i   -   v1f )   =   m2 ( v2f   -   v2i )             …………[3]
Rewrite the kinetic energy equations as
m1 ( v1i 2   -   v1f 2 )   =   m2 ( v2f 2   -   v2i 2 )     ………..[4]
[Noting that (a – b)(a + b) = a2 – b2], we write equation [4] as
m1 ( v1i   -   v1f )( v1i   -   v1f )   =   m2 ( v2f   -   v2i )( v2f   +   v2i )     ……….[5]
Divide equation [5] by equation [3], we get
v1i   +   v1f   =   v2f   +   v2i             ……….[6]
Rewrite this equation as
v1i   -   v2i   =   v2f   -   v1f   =   - ( v1f   -   v2f )   ……….[7]

Conservation of momentum   ( v2i   =   0)
m1 v1i   =   m1 v1f   +   m2 v2f           ……….[8]
Conservation of kinetic energy
½ m1 v1i 2   =   ½ m1 v1f   2   +   ½ m2 v2f   2     ………..[9]
From equation [6]
v2f   =   v1i   +   v1f
v1f   =   v2f   -   v1i         ……….[10]

Insert equation [10] into equation [8]
m1 v1i   =   m1 ( v2f   -   v1i )   +   m2 v2f
m1 v1i   =   m1 v2f   -   m1 v1i   +   m2 v2f
m1 v1i   +   m1 v1i   =   m1 v2f   +   m2 v2f
2 m1 v1i   =   ( m1   +   m2 ) v2f
[pic]             ……….[11]
and inserting   v2f   =   v1i   +   v1f   into the equation for conservation of momentum [8]
m1 v1i   =   m1 v1f   +   m2 ( v1f   +   v1i )
=   m1...