Goals: Lectue 13 Chapte 9 v Employ consevation of momentum in 1 D & 2D v Examine foces ove time (aka Impulse) Chapte 10 v Undestand the elationship between motion and enegy Assignments: l HW5, due tomoow l o Wednesday, Read all of Chapte 10 Physics 207: Lectue 13, Pg 1 Momentum is conseved if no extenal foce p befoe = p afte Physics 207: Lectue 13, Pg 2 Page 1
Inelastic collision in 1-D: Example l A block of mass M is initially at est on a fictionless hoizontal suface. A bullet of mass m is fied at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a final speed V. l In tems of m, M, and V : What is the momentum of the bullet with speed v? v V x befoe afte Physics 207: Lectue 13, Pg 3 Inelastic collision in 1-D: Example mv What is the momentum of the bullet with speed v? v Key question: Is thee a net extenal x-di foce? v If not, then momentum in the x-diection is conseved! P Befoe P Afte v mv + M 0 aaaa = ( m + M )V V befoe afte v= (1+M / m)v x Physics 207: Lectue 13, Pg 4 Page 2
p = p p p x y z = = mv = mv Momentum is a vecto mv mv = p x y z x iˆ + p y ˆj + p z kˆ Physics 207: Lectue 13, Pg 5 A pefectly inelastic collision in 2-D l Conside a collision in 2-D (cas cashing at a slippey intesection...no fiction). v 1 m 1 θ V m 1 + m 2 m 2 v 2 befoe afte l If no extenal foce momentum is conseved. l Momentum is a vecto so p x, p y and p z Physics 207: Lectue 13, Pg 6 Page 3
A pefectly inelastic collision in 2-D p 1 p V v 1 θ m 1 + m 2 m 1 m 2 v 2 p 2 befoe afte l x-di p x : m 1 v 1 = (m 1 + m 2 ) V cos θ l y-di p y : m 2 v 2 = (m 1 + m 2 ) V sin θ Physics 207: Lectue 13, Pg 7 Execise Momentum is a Vecto (!) quantity l A block slides down a fictionless amp and then falls and lands in a cat which then olls hoizontally without fiction l In egads to the block landing in the cat is momentum conseved? A. Yes B. No C. Yes & No D. Too little infomation given Physics 207: Lectue 13, Pg 8 Page 4
Execise Momentum is a Vecto (!) quantity l x-diection: No net foce so P x is conseved. l y-diection: Net foce, inteaction with the gound so depending on the system (i.e., do you include the Eath?) P y is not conseved (system is block and cat only) 2 kg 5.0 m Let a 2 kg block stat at est on a 30 incline and slide vetically a distance 5.0 m and fall a distance 7.5 m into the 10 kg cat 30 7.5 m 10 kg What is the final velocity of the cat? Physics 207: Lectue 13, Pg 9 Execise Momentum is a Vecto (!) quantity l x-diection: No net foce so P x is conseved l y-diection: v y of the cat + block will be zeo and we can ignoe v y of the block when it lands in the cat. j N i 5.0 m 30 mg 30 7.5 m 1) a i = g sin 30 = 5 m/s 2 2) d = 5 m / sin 30 = ½ a i t 2 10 m = 2.5 m/s 2 t 2 2s = t v = a i t = 10 m/s v x = v cos 30 = 8.7 m/s y x Physics 207: Lectue 13, Pg 10 Page 5
Execise Momentum is a Vecto (!) quantity l x-diection: No net foce so P x is conseved l y-diection: v y of the cat + block will be zeo and we can ignoe v y of the block when it lands in the cat. j N Initial inal i 5.0 m 30 mg 30 P x : MV x + mv x = (M+m) V x M 0 + mv x = (M+m) V x V x = m v x / (M + m) = 2 (8.7)/ 12 m/s V x = 1.4 m/s y 7.5 m x Physics 207: Lectue 13, Pg 11 Elastic Collisions l Elastic means that the objects do not stick. l Thee ae many moe possible outcomes but, if no extenal foce, then momentum will always be conseved l Stat with a 1-D poblem. Befoe Afte Physics 207: Lectue 13, Pg 12 Page 6
Billiads l Conside the case whee one ball is initially at est. befoe afte p a θ p b P a φ The final diection of the ed ball will depend on whee the balls hit. Physics 207: Lectue 13, Pg 13 Billiads: Without extenal foces, consevation of momentum (and enegy Ch. 10 & 11) l Consevation of Momentum l x-di P x : m v befoe = m v afte cos θ + m V afte cos φ l y-di P y : 0 = m v afte sin θ + m V afte sin φ befoe afte p afte θ p b P afte φ Physics 207: Lectue 13, Pg 14 Page 7
Home Execise Inelastic Collision in 1-D with numbes Do not ty this at home! ice (no fiction) Befoe: A 4000 kg bus, twice the mass of the ca, moving at 30 m/s impacts the ca at est. What is the final speed afte impact if they move togethe? Physics 207: Lectue 13, Pg 15 M = 2m Home execise Inelastic Collision in 1-D m initially MV V 0 M m + M ice v = 0 (no fiction) 2m 2m + m 0 = ( m + M)V o V = V0 = V0 finally 2V o /3 = 20 m/s v f =? Physics 207: Lectue 13, Pg 16 Page 8
Explosions: A collision in evese l A two piece assembly is hanging vetically at est at the end of a 2.0 m long massless sting. The mass of the two pieces ae 60 and 20 kg espectively. Suddenly you obseve that the 20 kg is ejected hoizontally at 30 m/s. The time of the explosion is shot compaed to the swing of the sting. l Does the tension in the sting incease o decease afte the explosion? Befoe Afte Physics 207: Lectue 13, Pg 17 Explosions: A collision in evese l A two piece assembly is hanging vetically at est at the end of a 2.0 m long massless sting. The mass of the two pieces ae 60 and 20 kg espectively. Suddenly you obseve that the 10 kg is ejected hoizontally at 30 m/s. l Deciphe the physics: 1. The geen ball ecoils in the x diection (3 d Law) and, because thee is no net foce in the x-diection the x- momentum is conseved. 2. The motion of the geen ball is constained to a cicula path thee must be centipetal (i.e., adial acceleation) Befoe Afte Physics 207: Lectue 13, Pg 18 Page 9
Explosions: A collision in evese l A two piece assembly is hanging vetically at est at the end of a 20.0 m long massless sting. The mass of the two pieces ae 60 & 20 kg espectively. Suddenly you obseve that the 20 kg mass is suddenly ejected hoizontally at 30 m/s. l Cons. of x-momentum P x befoe = P x afte = 0 = - M V + m v V = m v / M = 20*30/ 60 = 10 m/s T befoe = Weight = (60+20) x 10 N = 800 N Befoe Afte Σ y = m a y (adial) = M V 2 / = T Mg T = Mg + MV 2 / = 600 N + 60x(10) 2 /20 N = 900 N Physics 207: Lectue 13, Pg 19 Impulse (A vaiable foce applied fo a given time) l Gavity: At small displacements a constant foce l Spings often povide a linea foce (-k x) towads its equilibium position (Chapte 10) l Collisions often involve a vaying foce (t): 0 maximum 0 l We can plot foce vs time fo a typical collision. The impulse, J, of the foce is a vecto defined as the integal of the foce duing the time of the collision. Physics 207: Lectue 13, Pg 20 Page 10
oce and Impulse (A vaiable foce applied fo a given time) l J a vecto that eflects momentum tansfe J t t dt = ( dp / dt) dt = p dp Impulse J = aea unde this cuve! (Tansfe of momentum!) Impulse has units of Newton-seconds t i t t f t Physics 207: Lectue 13, Pg 21 oce and Impulse l Two diffeent collisions can have the same impulse since J depends only on the momentum tansfe, NOT the natue of the collision. same aea t t big, small t t t small, big t Physics 207: Lectue 13, Pg 22 Page 11
Aveage oce and Impulse av av t t big, av small t t t small, av big t Physics 207: Lectue 13, Pg 23 Execise oce & Impulse l Two boxes, one heavie than the othe, ae initially at est on a hoizontal fictionless suface. The same constant foce acts on each one fo exactly 1 second. Which box has the most momentum afte the foce acts? light heavy A. heavie B. lighte C. same D. can t tell Physics 207: Lectue 13, Pg 24 Page 12
Boxing: Use Momentum and Impulse to estimate g foce Physics 207: Lectue 13, Pg 25 Back of the envelope calculation J (1) m am ~ 7 kg (2) v am ~7 m/s (3) Impact time t ~ 0.01 s Question: Ae these easonable? Impulse J = p ~ m am v am ~ 49 kg m/s avg ~ J/ t ~ 4900 N (1) m head ~ 7 kg = dt a head = / m head ~ 700 m/s 2 ~ 70 g! t = t l Enough to cause unconsciousness ~ 40% of a fatal blow l Only a ough estimate! avg Physics 207: Lectue 13, Pg 26 Page 13
Discussion Execise l The only foce acting on a 2.0 kg object moving along the x- axis. Notice that the plot is foce vs time. l If the velocity v x is +2.0 m/s at 0 sec, what is v x at 4.0 s? l p = m v = Impulse l m v = J 0,1 + J 1,2 + J 2,4 l m v = (-8)1 N s + ½ (-8)1 N s + ½ 16(2) N s m v = 4 N s v = 2 m/s v x = 2 + 2 m/s = 4 m/s Physics 207: Lectue 13, Pg 27 Ch. 10 : Kinetic & Potential enegies l Kinetic enegy, K = ½ mv 2, is defined to be the lage scale collective motion of one o a set of masses l Potential enegy, U, is defined to be the hidden enegy in an object which, in pinciple, can be conveted back to kinetic enegy l Mechanical enegy, E Mech, is defined to be the sum of U and K. Physics 207: Lectue 13, Pg 28 Page 14
Lectue 13 Assignment: l HW6 up soon l o Wednesday: Read all of chapte 10 Physics 207: Lectue 13, Pg 29 Page 15