Physcs 05: Mechancs Lecture 3 Wenda Cao NJIT Physcs Department Momentum and Momentum Conseraton Momentum Impulse Conseraton o Momentum Collsons Lnear Momentum A new undamental quantty, lke orce, energy The lnear momentum p o an object o mass m mong wth a elocty s dened to be the product o the mass and elocty: p m The terms momentum and lnear momentum wll be used nterchangeably n the text Momentum depend on an object s mass and elocty
Lnear Momentum, cont Lnear momentum s a ector quantty p m Its drecton s the same as the drecton o the elocty The dmensons o momentum are ML/T The SI unts o momentum are kg m Momentum can be expressed n component orm: p x m x p y m y p z m z Newton s Law and Momentum Newton s Second Law can be used to relate the momentum o an object to the resultant orce actng on t Δ Δ( m) F net ma m Δ t The change n an object s momentum dded by the elapsed tme equals the constant net orce actng on the object Δp change n momentum F net tme nteral Impulse When a sngle, constant orce acts on the object, there s an mpulse delered to the object I F I s dened as the mpulse The equalty s true een the orce s not constant Vector quantty, the drecton s the same as the drecton o the orce Δp change n momentum F net tme nteral
Impulse-Momentum Theorem The theorem states that the mpulse actng on a system s equal to the change n momentum o the system Δp Fnet I I Δp m m How Good Are the Bumpers? In a crash test, a car o mass.5 0 3 kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are -5 m/s and.6 m/s, respectely. I the collson lasts or 0.5 s, nd (a) the mpulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car How Good Are the Bumpers? In a crash test, a car o mass.5 0 3 kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are -5 m/s and.6 m/s, respectely. I the collson lasts or 0.5 s, nd (a) the mpulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car 3 p m (.5 0 kg)( 5m ).5 0 kg m s p / 3 m (.5 0 kg)( +.6m ) + 0.39 0 kg m / I p p m m (0.39 0 kg m ) (.5 0 kg m ).6 0 kg m Δp I.6 0 kg m.76 0 N 0.5s F a 5 s 3
Impulse-Momentum Theorem A chld bounces a 00 g superball on the sdewalk. The elocty changes o the superball s rom 0 m/s downward to 0 m/s upward. I the contact tme wth the sdewalk s 0.s, what s the magntude o the mpulse mparted to the superball? (A) 0 (B) kgm/s (C) 00 kgm/s (D) 000 kgm/s (E) 0000 kgm/s I Δp m m Conseraton o Momentum In an solated and closed system, the total momentum o the system remans constant n tme. Isolated system: no external orces Closed system: no mass enters or leaes The lnear momentum o each colldng body may change The total momentum P o the system cannot change. Conseraton o Momentum Start rom mpulse-momentum theorem F m m m m F Snce Then So m F t F Δ m ) m ( m m + m m + m
Conseraton o Momentum When no external orces act on a system consstng o two objects that collde wth each other, the total momentum o the system remans constant n tme Fnet Δp p p When F then Δp net 0 0 For an solated system p p Speccally, the total momentum beore the collson wll equal the total momentum ater the collson + m m + m m The Archer An archer stands at rest on rctonless ce and res a 0.5-kg arrow horzontally at 50.0 m/s. The combned mass o the archer and bow s 60.0 kg. Wth what elocty does the archer moe across the ce ater rng the arrow? p p m + m m + m m 60.0kg, m 0.5kg, 0, 50m,? 0 + m m m 0.5kg (50.0m ) 0.7m m 60.0kg Conseraton o Momentum A 00 kg man and 50 kg woman on ce skates stand acng each other. I the woman pushes the man backwards so that hs nal speed s m/s, at what speed does she recol? (A) 0 (B) 0.5 m/s (C) m/s (D). m/s (E) m/s 5
Types o Collsons Momentum s consered n any collson Inelastc collsons: rubber ball and hard ball Knetc energy s not consered Perectly nelastc collsons occur when the objects stck together Elastc collsons: bllard ball both momentum and knetc energy are consered Actual collsons Most collsons all between elastc and perectly nelastc collsons Collsons Summary In an elastc collson, both momentum and knetc energy are consered In an nelastc collson, momentum s consered but knetc energy s not. Moreoer, the objects do not stck together In a perectly nelastc collson, momentum s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the same Elastc and perectly nelastc collsons are lmtng cases, most actual collsons all n between these two types Momentum s consered n all collsons Collson A car and a large truck traelng at the same speed collde head-on and stck together. Ths collson s (A) Inelastc collson (B) Perectly nelastc collson (C) Elastc collson (D) Impossble to determne 6
More about Perectly Inelastc Collsons When two objects stck together ater the collson, they hae undergone a perectly nelastc collson Conseraton o momentum m ) + m ( m + m m + m m + m Knetc energy s NOT consered An SUV Versus a Compact An SUV wth mass.80 0 3 kg s traellng eastbound at +5.0 m/s, whle a compact car wth mass 9.00 0 kg s traellng westbound at -5.0 m/s. The cars collde head-on, becomng entangled. (a) Fnd the speed o the entangled cars ater the collson. (b) Fnd the change n the elocty o each car. (c) Fnd the change n the knetc energy o the system consstng o both cars. An SUV Versus a Compact (a) Fnd the speed o the entangled cars ater the collson. p p 3 m.80 0 kg, + 5m m 9.00 0 kg, 5m m + m ( m m ) ( + m + m m + m +5.00m 7
An SUV Versus a Compact (b) Fnd the change n the elocty o each car. +5.00m Δ 0.0m Δ + 0.0m 3 m.80 0 kg, + 5m m 9.00 0 kg, 5m mδ m ( ).8 0 kg m m Δ m ( ) +.8 0 kg m s / m Δ + mδ 0 An SUV Versus a Compact (c) Fnd the change n the knetc energy o the system consstng o both cars. +5.00m 5 KE m + m 3.0 0 J KE m + m 3.38 0 J 5 ΔKE KE KE.70 0 J 3 m.80 0 kg, + 5m m 9.00 0 kg, 5m Collson and Momentum A car and a large truck traelng at the same speed collde head-on and stck together. Whch ehcle experences the larger change n the magntude o ts momentum? (A) the car (B) the truck (C) the change n the magntude o momentum s the same or both (D) Impossble to determne 8
More About Elastc Collsons Both momentum and knetc energy are consered m + m m + m m + m m + m Typcally hae two unknowns Momentum s a ector quantty Drecton s mportant Be sure to hae the correct sgns Sole the equatons smultaneously Elastc Collsons A smpler equaton can be used n place o the KE equaton m + m m + m m ( ) m( ) m + )( + ) (m ( )( ) ) ( m + m m + m m ( ) m( ) + + m + m m + m Summary o Types o Collsons In an elastc collson, both momentum and knetc energy are consered + + m + m m + m In an nelastc collson, momentum s consered but knetc energy s not + m m m m + In a perectly nelastc collson, momentum s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the same m ) + m ( m + m 9
Conseraton o Momentum An object o mass m moes to the rght wth a speed. It colldes head-on wth an object o mass 3m mong wth speed /3 n the opposte drecton. I the two objects stck together, what s the speed o the combned object, o mass m, ater the collson? (A) 0 (B) / (C) (D) (E) Problem Solng or D Collsons, Coordnates: Set up a coordnate axs and dene the eloctes wth respect to ths axs It s conenent to make your axs concde wth one o the ntal eloctes Dagram: In your sketch, draw all the elocty ectors and label the eloctes and the masses Problem Solng or D Collsons, Conseraton o Momentum: Wrte a general expresson or the total momentum o the system beore and ater the collson Equate the two total momentum expressons Fll n the known alues m + m m + m 0
Problem Solng or D Collsons, 3 Conseraton o Energy: I the collson s elastc, wrte a second equaton or conseraton o KE, or the alternate equaton Ths only apples to perectly elastc collsons + + Sole: the resultng equatons smultaneously Let s Play Pool Two bllard balls o dentcal mass moe toward each other. Assume that the collson between them s perectly elastc. I the ntal eloctes o the balls are +30.0 cm/s and -0.0 cm/s, what s the elocty o each ball ater the collson? Assume rcton and rotaton are unmportant. m m, + 30m m m, 0m m + m m + m + + Let s Play Pool Two bllard balls o dentcal mass moe toward each other. Assume that the collson between them s perectly elastc. I the ntal eloctes o the balls are +30.0 cm/s and -0.0 cm/s, what s the elocty o each ball ater the collson? Assume rcton and rotaton are unmportant. m m, +30 m / s + + m m, 0m m + m m + m 0.0 + cm 50.0 + 30.0cm 0.0cm cm