Impulse and Momentum Linear momentum impulse Linear momentum

Impulse and Momentum

Linear momentum & impulse • Linear momentum is defined as the product of mass and velocity – p=mv, px=mvx , py= mvy – units of momentum are kgm/s • From Newtons 2 nd law • F= ma F=m v/ t F= p/ t • The rate of momentum change with respect to time is equal to the resultant force on an object • The product of Force and time is known as IMPULSE • J= F t • units of impulse are Ns

Linear momentum & impulse Examples of impulses being applied on everyday objects

Impulse Momentum Theorem F t=m v You apply an impulse on an object and you get an equal change in momentum Area under a Force vs time graph

Impulse Graph

Conservation of momentum 2 particle system For gravitational or electrostatic force m 1 m 2 F 12 F 21 F 12 =dp 1/dt F 21 = dp 2/dt F 12 is force of 1 on 2 F 21 is force of 2 on 1

Conservation of momentum 2 particle system From Newton’s 3 rd Law F 12 = - F 21 m 1 or F 12 + F 21 = 0 m 2 F 12 F 21 F 12 + F 21 =dp 1/dt + dp 2/dt = 0 d(p 1 + p 2)/dt= 0 F 12 is force of 1 on 2 F 21 is force of 2 on 1 Since this derivative is equal to 0

Conservation of momentum 2 particle system Since this derivative is d(p 1 + p 2)/dt= 0 then integration yields equal to 0 p 1 + p 2 = a CONSTANT F 12 m 1 F 12 F 21 Thus the total momentum of the system of 2 particles is a constant. is force of 1 on 2 is force of 2 on 1

Conservation of linear momentum Provided the particles are isolated from external forces, the total momentum of the particles will remain constant regards of the interaction between them F 12 m 1 F 21 m 2 Simply stated: when two particles collide, their total momentum remains constant. pi = pf p 1 i + p 2 i = p 1 f + p 2 f (m 1 v 1)i + (m 2 v 2)i = (m 1 v 1)f + (m 2 v 2)f

Collisions

Collisions Event when two particles come together for a short time producing impulsive forces on each other. , No external forces acting. Or for the enthusiast: External forces are very small compared to the impulsive forces Types of collisions 1) Elastic- Momentum and Kinetic energy conserved 2) Inelastic- Momentum conserved, some KE lost 3) Perfectly(completely) Inelastic- Objects stick together

Collisions in 1 d Perfectly Elastic 1) Cons. of mom. 2) KE lost in collision 3) KE changes to PE

Elastic Collision Calculation 2 objects

Collisions - Examples Computer Simulations example 2, problems 5, 24, 29 Serway Problems 27, 29, 33, 37

Collisions in 2 dimensions After Collision x momentum before collision equals x momentum after the collision mavax Before collision mb vel=0 p=0 mavaf 1 mavafx mbvbxf mbvbf 2

Collisions in 2 dimensions mavax= mavafx + mbvbxf or mavax= mavaf cos 1 + mbvbf cos 2

Collisions in 2 dimensions After Collision y momentum before collision equals y momentum after the collision mavax Before collision Velocity y axis =0 py=o mavaf 1 mavayf mb vel=0 p=0 2 mbvbf Mbvbyf

Collisions in 2 dimensions 0= mavafy - mbvbfy or 0= mavaf sin 1 -mbvbf sin 2

Collisions in 2 dimensions 0= mavaf sin 1 -mbvbf sin 2 mavax= mavaf cos 1 + mbvbf cos 2