Physics 201 8 Linear Momentum and Collisions Linear

Physics 201 8: Linear Momentum and Collisions • Linear Momentum and its Conservation • Impulse and Momentum • Collisions • Elastic and Inelastic Collisions in One Dimension • Two Dimensional Collisions • The Center of Mass • Motion of a System of Particles • Rocket Propulsion

Linear Momentum and its Conservation Linear momentum p = mv Newtons Second law dv dp F tot = ma = m = dt dt (if the mass is constant º closed system ) dp =0 F tot = 0 Û dt p = CONSTANT Û Dp = 0

F 12 1 2 F 21

Impulse and Momentum dp = Fdt ò ò Þ dp = Dp = Fdt = I I is called the Impulse of the force I= ò F dt =area under force versus time gra Dp I F= = Dt Dt

Collisions Basic Assumption Forces during the collision dominate all other forces present, so only they need be considered This assumption implies that only contact forces between the objects are important , i. e. that the system is closed This in turn implies that the Total Linear Momentum of the system is conserved

Perfectly Inelastic Collisions The two particles stick together after the collision Þ v 1 f = v 2 f Thus using Dptot = 0 in one dimension m 1 v 1 i + m 2 v 2 i = m 1 v f +m 2 v f = v f (m 1 + m 2) m 1 v 1 i + m 2 v 2 i Þ vf = (m 1 + m 2 )

Elastic Collisions both Momentum ANDKinetic Energy are conserved m 1 v 1 i + m 2 v 2 i = m 1 v 1 f + m 2 v 2 f 1 1 m 1 v 1 i 2 + m 2 v 22 i = m 1 v 1 f 2 + m 2 v 22 f 2 2 éÞ 1 m v 2 - v 2 = 1 m v 2 - v 2 1 ( 1 i 1 f ) 2 ( 2 f 2 i ) 2 êÞ m 1 (v 1 i - v 1 f )(v 1 i + v 1 f ) = m 2 (v 2 f - v 2 i )(v 2 f êFrom conservation of total momentum(1) eq. ê m v - v 1 f ) = m 2 (v 2 f - v 2 i ) ê 1 ( 1 i ê (v 1 i + v 1 f ) = (v 2 f + v 2 i ) ë Þ v 1 i - v 2 i = - (v 1 f - v 2 f ) (1) (2 ) ù ú + v 2 i )ú ú ú û

Two Dimensional Collisions d=impact parameter

The Center of Mass In a system of two or more particles, the system moves as though all the mass were concentrated at a point. That point is called the center of mass. å 1 r = = mr å M m å åm (x i + y j + z k) 1 x i + y j+ z k = m (x i + y j + z k) = Må m k rk k =1, n cm k k k =1, n cm cm cm k k k =1, n k k k

Center of mass of extended objects 1 rcm = M ò 1 rdm= M object ò rr(r)dr; r = (x, y, z); ; dr = dxdydz object dm r(r) = ; M= d. V ò dm object c 1 xcm = M ò 1 xr(r)dr; ycm = M object ò 1 yr(r)dr; zcm = M object ò zr(r)dr object

Motion of a System of Particles

The effect of an external force is to change the momentum of the entire system. If the external force is zero the system maintains a zero or constant velocity and the total momentum of the system is conserved

Rocket Propulsion A rocket is a system with mass M moving at a velocityv. When the gases are expelled with exhaust velocity v e , then the rocket changes its velocity by d v. Thus through conservation of momentum M dv = - ved. M é M v i = ( M - d M )v f - d M v e ù ê M d v = - v d M if M ú d > > M ë û e Þ ò dv = - ve ò d. M M æ M ö i ÷ Þ D v = v e ln ç èMfø Thrust: F = M dv d. M = - ve dt dt