Phy 201: General Physics ILinear MomentumImpulse-Momentum TheoremNotes on ImpulseA Superman ProblemCollisionsConservation of Linear MomentumConservation of Momentum (Examples)Notes on Collisions & ForceCenter of MassRene Descartes (1596-1650)Phy 201: General Physics IChapter 7: Momentum & ImpulseLecture NotesLinear MomentumLinear momentum ( ) represents inertia in motion (Newton described momentum as the “quantity of motion”)Conceptually: reflects the effort required to bring a moving object to rest depends not only on its mass (inertia) but also on how fast it is movingDefinition: •Momentum is a vector quantity with the same direction as the object’s velocity•SI units are kg.m/sNewton’s 1st Law revisited:The momentum of an object will remain constant unless it is acted upon by a net force (or impulse)p = mvrrprImpulse-Momentum TheoremIn general, Newton’s 2nd Law, can be rewritten as Rearranging terms:this is called the Net (or average) Impulse!!Definition of Impulse associated with an applied force:•The SI units for impulse are N.s•Impulse represents simultaneously:1. The product of the force times the time:2. The change in linear momentum of the object: ( ){ } this is the simple casenet net where m is constant! mvp vF = = Note: F = m = ma t t tDD D�D D Drrrr rrnet net avgp = F t when F is not constant: p = F tD D � D Dr r rr ravgJ = p = F tD Dr rravgJ = F tDr rfJ = p = mv - mvDrrr rNotes on Impulse•Impulses always occur as action-reaction pairs (according to Newton’s 3rd Law)•The force.time relationship is observed in many “real world” examples:–Automobile safety:•Dashboards•Airbags•Crumple zones–Product packaging•Styrofoam spacers–Sports•Tennis: racket string tension•Baseball: “juiced” baseballs & baseball bats (corked & aluminum vs. wood)•Golf: the “spring-like” effect of golf club heads•Boxing gloves: (lower impulsive forces in the hands)A Superman ProblemIt is well known that bullets and missiles bounce off Superman’s chest. Suppose a bad guy sprays Superman’s chest with 0.003 kg bullets traveling at a speed of 300 m/s (fired from a machine gun at a rate of 100 rounds/min). Each bullet bounces straight back with no loss in speed. Problems:a) What is the impulse exerted on Superman’s chest by a single bullet?b) What is the average force exerted by the stream of bullets on Superman’s chest?CollisionsA specific type of interaction between 2 objects. The basic assumptions of a collision:1. Interaction is short lived compared to the time of observation2. A relatively large force acts on each colliding object3. The motion of one or both objects changes abruptly following collision4. There is a clean separation between the state of the objects before collision vs. after collision3 classifications for collisions:–Perfectly elastic: colliding objects bounce off each other and no energy is lost due to heat formation or deformation (Ksystem is conserved)–Perfectly inelastic: colliding objects stick together (Ksystem is not conserved)–Somewhat inelastic (basically all other type of collisions): KE is not conserved121ipr2ipr121fpr2fpr12121ipr2ipr1f+2fprConservation of Linear MomentumThe total linear momentum of a system will remain constant when no external net force acts upon the system, or•Note: Individual momentum vectors may change due to collisions, etc. but the linear momentum for the system remains constant•Useful for solving collision problems:–Where all information is not known/given–To simplify the problem•Conservation of Momentum is even more fundamental than Newton’s Laws!!1 2 before collision 1 2 after collision(p + p + ...) = (p + p + ...)r r r rConservation of Momentum (Examples)•The ballistic pendulum•2 body collisions (we can’t solve 3-body systems…)–Perfectly inelastic (Epre-collision ≠ Epost-collision)–Perfectly elastic (Epre-collision = Epost-collision)•Collisions in 2-D or 3-D:–Linear momentum is conserved by components:By Components:ˆ ˆ1x 2x 1x 2xbefore collision after collision(p + p + ...) i = (p + p + ...) i� � � �� � � �1 2 before collision 1 2 after collision(p + p + ...) = (p + p + ...)r r r rˆ ˆ1y 2y 1y 2ybefore collision after collision(p + p + ...) j = (p + p + ...)j� � � �� � � �Notes on Collisions & Force•During collisions, the forces generated:–Are short in duration–Are called impulsive forces (or impact forces or collision forces)–Often vary in intensity/magnitude during the event–Can be described by an average collision force:Example: a golf club collides with a 0.1 kg golf ball (initially at rest), t 0.01s. The velocity of the ball following the impact is 25 m/s.The impulse exerted on the ball is:The average impulsive force exerted on the ball is:The average impulsive force exerted on the club is:. .Net avgp impulseF = F = t timei eD� �� �D�rr rˆ ˆm ms sp = m v = (0.1 kg)(25 - 0 ) i = 2.5 N s iD D �rrˆ ˆavgp 2.5 N sF = = i = 250 N it 0.01 sD �Drrˆ ˆavgp -2.5 N sF = = i = -250 N it 0.01 sD �DrrCenter of MassCenter of Mass ( ) refers to the average location of mass for a defined mass.•To determine the center of mass, take the sum of each mass multiplied by its position vector and divide by the total mass of the system or•Note, if the objects in the system are in motion, the velocity of the system (center of mass) is: •When psystem = 0 (i.e. Fext = 0) then vcm = constant–The motion of all bodies even if they are changing individually will always have values such that vcm = constantcmrrni i1 1 2 2 3 3 n ni=1cm1 2 3 n sysmrm r + m r + m r + ... + m rr = = m + m + m + ... + m m�rr r r rrni i1 1 2 2 n n i=1cm1 2 n sysmvm v + m v + ... + m vv = = m m +...+ m m+�rr r rrRene Descartes (1596-1650)•Prominent French mathematician & philosopher•Active toward end of Galileo’s career•Studied the nature of collisions between objects•First introduced the concept of momentum (called it “vis-à-vis”) –he defined “vis-à-vis” as the product of weight times speed•Demonstrated the Law of Conservation of Momentum“Each problem that I solved became a rule which served afterwards to solve other