Linear Momentum. calculate the momentum of an object solve problems involving the conservation of momentum. Labs, Activities & Demonstrations:

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1 Add Important Linear Momentum Page: 369 Note/Cue Here NGSS Standard: HS-PS2-2 Linear Momentum MA Curriculum Framework (2006): 2.5 AP Phyic 1 Learning Objective: 3.D.1.1, 3.D.2.1, 3.D.2.2, 3.D.2.3, 3.D.2.4, 4.B.1.1, 4.B.1.2, 4.B.2.1, 4.B.2.2, 5.A.2.1, 5.D.1.1, 5.D.1.3, 5.D.1.4, 5.D.1.5, 5.D.2.2, 5.D.2.3, 5.D.2.4, 5.D.2.5, 5.D.3.1 Knowledge/Undertanding Goal: Skill: deinition o momentum calculate the momentum o an object olve problem involving the conervation o momentum Language Objective: Undertand and correctly ue the term colliion, elatic colliion, inelatic colliion, inertia and momentum. Accurately decribe and apply the concept decribed in thi ection uing appropriate academic language. Set up and olve word problem involving colliion and momentum. Lab, Activitie & Demontration: Note: Colliion on air track. Happy and ad ball knocking over a board. Student riding momentum cart. colliion: when two or more object come together and hit each other. elatic colliion: a colliion in which the object bounce o each other ater they collide, without any lo o kinetic energy. inelatic colliion: a colliion in which the object remain together ater colliding. In an inelatic colliion, total energy i till conerved, but ome o the energy i changed into other orm, o the amount o kinetic energy i dierent beore v. ater the colliion. Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

2 Add Important Linear Momentum Page: 370 Note/Cue Here Any macrocopic colliion (meaning a colliion between object that are larger than individual atom or molecule) will convert ome o the kinetic energy into internal energy and other orm o energy. Thi mean that no large cale impact are ever perectly elatic. In the 17 th century, the German mathematician Gottried Leibnitz recognized the act that in ome cae, the ma and velocity o object beore and ater a 2 colliion were related by kinetic energy ( 1 mv, which he called the quantity o motion ); in other cae, however, the quantity o motion wa not preerved but another quantity (mv, which he called the motive orce ) wa the ame beore and ater. Debate about quantity o motion and motive orce continued through the 17 th and 18 th centurie. We now realize that the two quantitie are dierent, but related. Quantity o motion i what we now call kinetic energy ; motive orce i what we now call momentum. While total energy i alway conerved in a colliion, kinetic energy i not; during the colliion, kinetic energy may be converted to other orm o energy, uch a heat. Momentum, however, i alway conerved in a colliion, regardle o what happen to the energy. momentum ( p ): the amount o orce that a moving object could traner in a given time in a colliion. (Formerly called motive orce.) Momentum i a vector quantity given by the ormula: and i meaured in unit o N, or Note that an object at ret ( v 0) Ue thi pace or ummary and/or additional note. 2 p mv kgm. ha a momentum o zero. Momentum i the quantity that i tranerred in all colliion. Any problem that involve one or more colliion i a momentum problem. The net orce on an object i it change in momentum with repect to time: p dp Fnet t dt AP Phyic 1: Algebra-Baed

3 Add Important Linear Momentum Page: 371 Note/Cue Here inertia: an object ability to reit the action o a orce. Inertia and momentum are related, but are not the ame thing; an object ha inertia even at ret. An object momentum change i either it ma or it velocity change, but an the inertia o an object can change only i it ma change. Conervation o Momentum In a cloed ytem, momentum i conerved. Thi mean that unle there i an outide orce, the combined momentum o object ater they collide i equal to the combined momentum o the object beore the colliion. elatic colliion: a colliion in which the object are eparate both beore and ater the colliion. In the ollowing momentum bar chart, imagine that two object are moving in oppoite direction and then collide. Beore the colliion, the irt object ha a momentum o +3 arbitrary unit, and the econd ha a momentum o 1. The total momentum i thereore +3 + ( 1) = +2. Ater the colliion, the irt object ha a momentum o +1.5 and the econd ha a momentum o Becaue there are no orce changing the momentum o the ytem, the inal momentum mut alo be +2. h Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

4 Add Important Linear Momentum Page: 372 Note/Cue Here inelatic colliion: a colliion in which the object are joined (or are a ingle object) either beore or ater the colliion. Example o inelatic colliion include a peron catching a ball and a bullet being ired rom a gun. (The bullet and gun are together initially and eparate ater the colliion.) Solving Momentum Problem Almot all momentum problem involve the conervation o momentum law: beore ater p i p The ymbol i the Greek capital letter igma. In mathematic, the ymbol mean ummation. p mean the um o the momentum. The ubcript i mean initial (beore the colliion), and the ubcript mean inal (ater the colliion). In plain Englih, p mean ind each individual value o p (poitive or negative, depending on the direction) and then add them all up to ind the total. The conervation o momentum law mean that the total beore a colliion mut be equal to the total ater. Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

5 Add Important Linear Momentum Page: 373 Note/Cue Here For example, i you had a momentum problem with two object, the law o conervation o momentum become: beore p ater 1, i p2, i p1, p2, Notice that we have two ubcript ater each p, becaue we have two eparate thing to keep track o. The i and mean initial and inal, and the 1 and 2 mean object #1 and object #2. Becaue p mv, we can replace each p with m v. For our momentum problem with two object, thi become: beore m 1v1, i m2v2, i m1v1, m2v2, ater Note that there are ix eparate quantitie in thi problem: m 1, m 2, v 1, i, v 2, i, v 1,, and v 2,. A typical momentum problem will give you (or enable you to calculate) ive o thee, and will ak you or the ixth. Note alo that mot momentum problem do not mention the word momentum. The problem uually give inormation about mae and velocitie beore and ater ome ort o colliion, and it i up to you to realize that the problem involve conervation o momentum. Mot momentum problem involve colliion. Uually, there are two object initially, and the object either bounce o each other (elatic colliion) or tick together (inelatic colliion). For an elatic colliion between two object, the problem i exactly a decribed above: there are ix quantitie to conider: the two mae, the two initial velocitie, and the two inal velocitie. The equation relating them i: beore m 1v1, i m2v2, i m1v1, m2v2, ater To olve the problem, you need to obtain the quantitie given in the word problem and olve or the miing one. Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

6 Add Important Linear Momentum Page: 374 Note/Cue Here For an inelatic colliion between two object, the object tick together ater the colliion, which mean there i only one object aterward. The total ma o the object i m T m 1 m2, and there i only one object with a inal velocity. there are ive quantitie: the two mae, the two initial velocitie, and the inal velocity o the combined object. The equation relating them i: beore m v m v 1 1, i 2 2, i ater m v To olve thi problem, you need to obtain the quantitie given in the word problem, add the two mae to ind m T, and olve or the miing quantity. T Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

7 Add Important Linear Momentum Page: 375 Note/Cue Here Sample Problem Q: What i the momentum o a 15 kg object moving at a velocity o 3.0 m? A: p mv (15 kg)( 3.0 m p ) 45 kgm 45N The anwer i given a +45 N becaue momentum i a vector, and we indicate the direction a poitive or negative. Q: An object with a ma o 8.0 kg moving with a velocity o 5.0 m collide with a tationary object with a ma o 12 kg. I the two object tick together ater the colliion, what i their velocity? A: The momentum o the moving object beore the colliion i: p m v (8.0 )( 5.0) 40 N The tationary object ha a momentum o zero, o the total momentum o the two object combined i +40 N. Ater the colliion, the total ma i 8.0 kg + 12 kg = 20 kg. The momentum ater the colliion mut till be +40 N, which mean the velocity i: p m v 40 20v v Uing the equation, we would olve thi a ollow: beore ater p1, i p2, i p m1v1, i m2v2, i mt v (8)(5) (12)(0) (8 12) v v Ue thi pace or ummary and/or additional note. v m 2 m AP Phyic 1: Algebra-Baed

8 Add Important Linear Momentum Page: 376 Note/Cue Here Q: Mr. Stretchy ha a ma o 60. kg and i holding a 5.0 kg box a he ride on a kateboard toward the wet at a peed o 3.0. (Aume the 60. kg i the ma o Mr. Stretchy and the kateboard combined.) He throw the box behind him, giving it a velocity o 2.0 to the eat. m m What i Mr. Stretchy velocity ater throwing the box? A: Thi problem i like an inelatic colliion in revere; Mr. Stretchy and the box are together beore the colliion and apart aterward. The equation would thereore look like thi: v m v m m T i, bvb, Where the ubcript i or Mr. Stretchy, and the ubcript b i or the box. Note that ater Mr. Stretchy throw the box, he i moving one direction and the box i moving the other, which mean we need to be careul about our ign. Let chooe the direction Mr. Stretchy i moving (wet) to be poitive. Becaue the box i thrown to the eat, thi mean the inal velocity o the box will be v b, 2.0 m Plugging value rom the problem into the equation or the law o conervation o momentum, we get: beore ater pi p, pb, mt vi mv, mbv (60 5)( 3) 60 v, (5)( 2) v, ( 10) v v,, b, m Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

9 Add Important Linear Momentum Page: 377 Note/Cue Here m Q: A occer ball that ha a ma o 0.43 kg i rolling eat with a velocity o 5.0. It collide with a volleyball that ha a ma o 0.27 that i rolling wet with a velocity o 6.5 m. Ater the colliion, the occer ball i rolling to the wet with a velocity o Auming the colliion i perectly elatic and riction m between both ball and the ground i negligible, what i the velocity (magnitude and direction) o the volleyball ater the colliion? A: Thi i an elatic colliion, o the occer ball and the volleyball are eparate both beore and ater the colliion. The equation i: v m v m v m m, i v v, i, vvv, Where the ubcript i or the occer ball and the ubcript v i or the volleyball. In all elatic colliion, aume we need to keep track o the direction, which mean we need to be careul about our ign. We don t know which direction the volleyball will be moving ater the colliion (though a good gue would be that it will probably bounce o the occer ball and move to the eat). So let u arbitrarily chooe eat to be poitive and wet to be negative. Thi mean: quantity direction value initial velocity o occer ball eat 5.0 m initial velocity o volleyball wet 6.5 m inal velocity o occer ball wet 3.87 m Plugging value rom the problem into the equation or the law o conervation o momentum, we get: beore ater p, i pv, i p, pv, mv, i mvvv, i mv, mvvv, (0.43)(5.0) (0.27)( 6.5) (0.43)( 3.87) (0.27) v 2.15 ( 1.755) vv, vv, v, v, Ue thi pace or ummary and/or additional note. v m v, or 7.63 m to theeat. AP Phyic 1: Algebra-Baed

10 Add Important Linear Momentum Page: 378 Note/Cue Here Homework Problem 1. A large 10. kg port ball i caught by a 70. kg tudent on the track team. I the ball wa moving at 4.0 and the tudent wa tationary beore m catching the ball, how at will the tudent be moving ater catching the ball? Anwer: 0.5 m Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

11 Add Important Linear Momentum Page: 379 Note/Cue Here 2. An kg bullet i ired down the barrel o a gun, which i pointed to the let. The bullet accelerate rom ret to a velocity o What velocity doe the 1.5 kg gun acquire a a reult o thi impule? (Note that neither the bullet nor the gun wa moving beore the colliion. ) m Anwer: 2.7 to the right m Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

12 Add Important Linear Momentum Page: 380 Note/Cue Here Although Blank momentum bar chart are not provided or the remaining problem, eel ree to draw them i you would ind them helpul. 3. A 730 kg Mini run into a tationary kg port utility vehicle. I the Mini wa moving at 10. initially, how at will it be moving ater m making a completely inelatic colliion with the SUV? Anwer: 2.3 m 4. A 6.0 kg bowling ball moving at 3.5 m to the right make a colliion, head- on, with a tationary 0.70 kg bowling pin. I the ball i moving 2.77 to the right ater the colliion, what will be the velocity (magnitude and direction) o the pin? m Anwer: 6.25 to the right m 5. A pair o 0.20 kg billiard ball make an elatic colliion. Beore the colliion, the 4-ball wa moving 0.50 to the right, and the 8-ball wa moving 1.0 to the let. Ater the colliion, the 4-ball i now moving at m 1 to the let. What i the velocity (magnitude and direction) o the 8-.0 m ball ater the colliion? m Anwer: 0.50 to the right m Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed

13 Add Important Linear Momentum Page: 381 Note/Cue Here 6. A pair o billiard ball, each with ma m, make an elatic colliion. Beore the colliion, the 4-ball wa moving with a velocity o v 4, i, and the 8-ball wa moving with a velocity o v 8, i. Ater the colliion, the 4-ball i now moving with a velocity o v 4,. What i the velocity o the 8-ball ater the colliion? (You may ue your work rom problem #5 above to guide your algebra.) Anwer: v 8, v4, i v8, i v4, 7. A 75 kg atronaut on a pace walk puhe to the right on a kg atellite. I the velocity o the atellite ater the puh i 0.75, what i the velocity o the atronaut? m Anwer: 10 m to the let Ue thi pace or ummary and/or additional note. AP Phyic 1: Algebra-Baed