Phy 201: General Physics I Chapter 7: Momentum & Impulse Lecture Notes Linear Momentum Linear momentum ( p ) 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 moving Definition: p = mv Momentum is a vector quantity with the same direction as the objects velocity SI units are kg.m/s Newtons 1st Law revisited: The momentum of an object will remain constant unless it is acted upon by a net force (or impulse) Impulse-Momentum Theorem nd Law, can be rewritten as In general, Newtons 2
mv p v Fnet = = Note: Fnet = m = ma t t t this is the simple case where m is constant! Rearranging terms: p = Fnet t when Fnet is not constant: p = Favgt this is called the Net (or average) Impulse!! Definition of Impulse
associated with an applied force: J = p = Favgt The SI units for impulse are N.s Impulse represents simultaneously: 1. The product of the force times the time: J = Favgt 2. The change in linear momentum of the object: J =p = mv fi- mv Notes on Impulse Impulses always occur as action-reaction pairs (according to Newtons 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 Problem It is well known that bullets and missiles bounce off Supermans chest. Suppose a bad guy sprays Supermans 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 Supermans chest by a single bullet? b) What is the average force exerted by the stream of bullets on Supermans chest? Collisions A specific type of interaction between 2 objects. The basic assumptions of a collision: 1. Interaction is short lived compared to the time of observation 2. A relatively large force acts on each colliding object 3. The motion of one or both objects changes abruptly following collision 4. There is a clean separation between the state of the objects before collision vs. after collision
3 classifications for collisions: Perfectly elastic: colliding objects bounce off each other and no energy is lost due to heat formation or deformation (K system is conserved) 1 2 1 2 1 2 p p2f p1i p2i 1f Perfectly inelastic: colliding objects stick together (Ksystem is not conserved) 1 2
p1i p2i p1f+2f Somewhat inelastic (basically all other type of collisions): KE is not conserved Conservation of Linear Momentum The total linear momentum of a system will remain constant when no external net force acts upon the system, or (p1 + p2 + ...)before collision= (p1 + p2 + ...) after collision 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 Newtons Laws!! Conservation of Momentum (Examples) The ballistic pendulum 2 body collisions (we cant 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: (p1 + p2 + ...)before collision= (p1 + p2 + ...)after collision By Components: (p1x + p2x + ...) i = (p1x + p2x + ...) i before collision after collision (p1y + p2y + ...) j = (p1y + p2y + ...)j before collision after collision 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:
p impulse FNet = Favg = i . e . t time 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: p = mv = (0.1 kg)(25 m - 0 m ) i = 2.5 N s i s s The average impulsive force exerted on the ball is: p 2.5 N s Favg = =
i = 250 N i t 0.01 s The average impulsive force exerted on the club is: p -2.5 N s Favg = = i = -250 N i t 0.01 s Center of Mass Center of Mass ( rcm) 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 n mr
i i m1r1 + m2r2 + m3r3 + ... + mnrn rcm = = i=1 m1 + m2+ m3 + ... + mn msys Note, if the objects in the system are in motion, the velocity of the system (center of mass) is: n m v + m2v2+ ... + mnvn vcm = 1 1 = m1 m2+...+ mn mv i i i=1 msys 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 = constant
Rene Descartes (1596-1650) Prominent French mathematician & philosopher Active toward end of Galileos career Studied the nature of collisions between objects First introduced the concept of momentum (called it vis--vis) he defined vis--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 problems.
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