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 v1f + m2 v1f + m2 v1i m1 v1i - m2 v1i = ( m1 + m2 ) v1f [pic] ……….[12]