Class 12: Coefficient of restitution and Class 12: Coefficient of - PowerPoint PPT Presentation

Class 12: Coefficient of restitution and Class 12: Coefficient of restitution and elastic collision

Center of mass Many particles m 1 , m 2 , m 3 , …. located at (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ),…. respectively. ∑ ∑ ⎫ ⎫ m x + + + i i L ⎪ m x m x m x = = 1 1 2 2 3 3 i x v ∑ ⎪ ∑ + + + cm L m m m m v v v m r ⎪ 1 2 3 i + + + i i v L m r m r m r ⇒ = = i ⎬ ⎬ 1 1 2 2 3 3 i r ∑ ∑ ∑ ∑ + + + cm cm L ⎪ ⎪ m m m m m y + + + 1 2 3 i i i L m y m y m y ⎪ = = i 1 1 2 2 3 3 i y ∑ + + + cm ⎪ L m m m m 1 2 3 i ⎭ i ∑ ⎫ m v + + + L m v m v m v i x ⎪ i = = 1 x 2 x 3 x i v 1 2 3 ∑ ∑ v ⎪ ∑ ∑ + + + + + + x L cm m m m m m m m m v v v m m v v ⎪ ⎪ + + + 1 1 2 2 3 3 i i i i L v m v m v m v ⇒ = = i ⎬ 1 1 2 2 3 3 i v ∑ ∑ + + + cm L m m m m ⎪ m v + + + L 1 2 3 i m v m v m v i y i ⎪ = = 1 y 2 y 3 y i i v 1 2 3 ∑ ∑ + + + + + + ⎪ ⎪ y L L cm m m m m m m m m 1 2 3 i ⎭ i

Physics of the center of mass 1. Momentum: Total momentum of all particles observed in the center of mass frame = 0. 2. Kinetic energy: 1 = CM + 2 Total Kinetic Energy Mv Kinetic energy observed in the CM frame 2 1 = ∑ − 2 Kinetic energy observed in the CM frame m ( v v ) i i CM 2 i M = Total mass of all particles, v CM = velocity of the center of mass v v Total = 3. F M a CM v F is the total external force acting on all particles. Total = M Total mass of all particles.

C Coefficient of restitution ffi i t f tit ti − v v ' ' v v ' ' = 2 1 e - − v v 2 1 relative velocity after collision v' = = rel - - relative velocity y before collision v rel rel 0 ≤ e ≤ 1: 0 ≤ e ≤ 1: e=1 for elastic collision e=0 for completely inelastic collision e 0 for completely inelastic collision

El Elastic collision ti lli i + = + m v m v m v ' m v ' 1 1 2 2 1 1 2 2 1 1 1 1 1 1 1 1 + = + 2 2 2 2 m v m v m v ' m v ' 1 1 2 2 1 1 2 2 2 2 2 2 Replace − v ' v ' = 2 1 1 - − v v 2 1

General collision problem General collision problem + + = ' + + m v m v m v ' m v ' ' (1) (1) 1 1 2 2 1 1 2 2 − v ' v ' = 2 1 e - (2) − v v v v 2 2 1 1 Two equations two unknown (v ’ and v ’) Two equations, two unknown (v 1 and v 2 )

Energy loss in a collision Energy loss in a collision 1 1 m m m m = 2 2 1 2 Energy Loss (v - v ) (1 - e ) + 2 1 2 m m 1 2 1 = μ 2 2 v (1 - e ) rel 2 m m 1 m m μ = 1 2 2 reduced mass + m m 1 2