Physics 30 Lesson 1 Momentum and Conservation of Momentum in One Dimension

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1 Phyic 30 Leon 1 Moentu and Conervation of Moentu in One Dienion I. Phyic rincile Student often ak e if Phyic 30 i harder than Phyic 0. Thi, of coure, deend on the atitude, attitude and work ethic of the individual tudent. However, there i one ajor difference between Phyic 0 and Phyic 30. Phyic 0 wa doinated by roble olving and the calculation of an anwer. Phyic 30 ha a ubtantial roble-olving coonent, but it alo require that tudent learn and undertand the Phyic Princile that for the foundation of hyic. (Thee rincile are lited on your Phyic Data Sheet and are reroduced to the right.) In other word, you will be required to exlain how the hyic rincile are being alied to a articular roble you will deontrate that you know the theory behind the roble olving. Soe of thee rincile (0, 1,, 3 and 5) were taught in Phyic 0 and we will ee the again in different context in Phyic 30. The reaining rincile are what Phyic 30 i all about. Thi leon will introduce rincile 4, the conervation of oentu. 0 Unifor otion ( F 0) net 1 Accelerated otion ( F 0) Unifor circular otion ( i radially inward) net F 3 Work-energy theore 4 Conervation of oentu 5 Conervation of energy 6 Conervation of a-energy 7 Conervation of charge 8 Conervation of nucleon 9 Wave-article duality net Dr. Ron Licht 1 1

2 II. Moentu A very ueful hyical concet i oentu. The oentu ( ) of an object i defined a the roduct of it a and velocity. Recall that velocity i a vector quantity that involve both a eed and a direction. Thu, oentu i a vector quantity. v a v velocity oentu There i no choen unit for oentu like there i for force (N) or energy (J). The unit for oentu i a cobination of the a unit and velocity unit. Soe exale are kg /, kg k/h, g /, etc. (You ay refer to Pearon age 446 to 449 for a different dicuion about oentu.) Exale 1 What i the oentu of a 1500 kg car travelling wet at 5.0 /? v 1500kg(5.0 wet) wet 3 kg Exale A icro-eteorite with a a of 5.0 g ha the ae oentu a the car in Exale 1. What i the velocity of the eteorite? v v v kg 3 kg v wet 6 wet A ajor iconcetion that eole have i that inertia and oentu ean the ae thing. In the exale above, both object have the ae oentu (a x velocity) but the inertia (i.e. a) of the car i ubtantially larger than the inertia of the eteorite. In other word, inertia refer to the a of an object while oentu refer to the a and otion of the object. Dr. Ron Licht 1

3 III. Syte Before we can undertand oentu ore fully, we ut alo be aware of different kind of yte. There are everal tye of yte: Cloed yte: no a enter or leave the yte Iolated yte: no external force act on the yte and no energy leave the yte. Oen yte: a ay enter or leave the yte and external force ay influence the yte. To illutrate the difference between thee kind of yte, conider a real-life colliion between two car: Firt, the colliion of the car i not an iolated yte i.e. they are not iolated fro the Earth. Frictional force between the Earth and the car will caue the car to low down. If the car collided on a very liery urface where the frictional force with the Earth were iniied, thi would alot contitute an iolated yte. Second, if the car collide and all of the art of the car tay attached to the car we have a cloed yte. However, if art of the car fly off we have an oen yte. In our invetigation of the conervation of oentu below we will be auing cloed and iolated yte. IV. Conervation of Moentu The real iortance of the concet of oentu i that in any iolated cloed yte, the total oentu of the yte i conerved (i.e. reain contant). For exale, conider two object ( 1 and ) that collide a hown below. v 1 1 v Before colliion During colliion v 1 1 v After colliion Although the oentu of each individual object change during the colliion, the u of the oenta before the colliion (1v1 v ) and after the colliion (1v1 v ) are the ae. The general tateent for the law of conervation of oentu i: The total oentu of an iolated yte of object reain contant. Dr. Ron Licht 1 3

4 In other word, the u of the oenta before a colliion or exloion ( ) equal the u of the oenta after a colliion or exloion ( ). In any colliion or exloion, the total oentu i alway conerved. Thi rincile rove to be very ueful in redicting what will haen when object collide or exlode. Actually, the rincile of the Conervation of Moentu i a direct conequence of Newton Third Law of Motion that we learned about in Phyic 0. Recall that when any object exert a force on another object, the econd object will exert an equal and ooite force on the firt object. Conider the colliion between the two ae illutrated above. When they collide, a 1 exert a force on a ( F 1on ) reulting in an acceleration (change in velocity) of a. According to Newton 3 rd Law, if object 1 exert a force on object, object will exert an equal and ooite reaction force on object 1 ( F on1 ) reulting in a change in velocity of a. Therefore F F 1on on1 If we aly Newton Second Law of Motion F 1on v t F on1 v t 1 1 v 1v 1 v F a t Notice that when the change in oentu of 1 and are added together, the reult i zero. v 1v 1 0 F 1on on1 Note that the rie ybol i ued to denote after the colliion or exloion. caue a change in the oentu of, and F caue a change in oentu of object 1 the tie of contact i the ae for both object t cancel out Thi reult indicate that while the oentu of each object change, the total change in oentu of the yte (i.e. both object) i zero. Thi i alo a tateent of the rincile of conervation of oentu. Dr. Ron Licht 1 4

5 V. Elatic and inelatic colliion There are two baic tye of colliion elatic and inelatic and we ue the law of conervation of oentu to redict and calculate the reult. Elatic colliion are thoe where both oentu and kinetic energy are conerved. Purely elatic colliion are very hard to roduce in the ordinary world becaue there i alway oe kinetic energy converted into heat, ound, deforation or oe other for of energy. A we hall ee in future leon on the nature of the ato, colliion between ubatoic article are elatic colliion. Inelatic colliion are thoe where oentu i conerved, but kinetic energy i not conerved. A coletely inelatic colliion i when the object collide and tick together. Exale 3 A.00 kg object travelling eat at 4.00 / collide with a 3.00 kg object travelling wet at 1.50 /. If the 3.00 kg object end u travelling eat at 1.5 /, what i the final velocity of the.00 kg object? Before After v 1 = / 1 =.00 kg v v v v v1 v v v kg( 4.00 ) 3.00kg( 1.50 ) 3.00kg( 1.5 ) v 1.00kg v 0.13 or 0.13 wet 1 v = 1.50 / = 3.00 kg v1? 1 =.00 kg v = +1.5 / = 3.00 kg When we aly the conervation of oentu rincile we want the equation to reflect the context of the quetion. In thi quetion there are two object before the colliion and two object after the colliion. Therefore there are two v ter on the before ide and two v ter on the after ide. Rearrange the equation, lug in the known value, and calculate the unknown. Dr. Ron Licht 1 5

6 Exale 4 A 30 kg object travelling at 45 / wet collide with a 40 kg object at ret. If the object tick together on contact, what i the reulting velocity of the cobined ae? Before After = 40 kg v v v v v v 1 = -45 / v v 1 = 30 kg 30kg( 45 ) 0 40kg 30kg v 19 or 19 wet v? 1+ = 40 kg + 30 kg In thi quetion there are two object before the colliion and one object after the colliion. Therefore there are two v ter on the before ide and one v ter on the after ide. Thi i an exale of a coletely inelatic colliion. Exale 5 A 400 kg object travelling eat at 50 / collide with a oving 50 kg object. The ae tick together after they collide and ove at 7.8 / to the eat. What wa the initial velocity of the 50 kg object? Before After v 1 = + 50 / v =? v = / 1 = 400 kg v v v v v v v (400kg 50kg)( 7.8 ) 400kg( 50 ) 50kg v wet = 50 kg 1+ = 400 kg + 50 kg Dr. Ron Licht 1 6

7 Exale 6 A.00 kg object travelling eat at 4.00 / collide with a 3.00 kg object travelling wet at 1.50 /. If the 3.00 kg object end u travelling eat at.10 /. a. What i the final velocity of the.00 kg object? b. Wa the colliion elatic? c. How can oentu alway be conerved while kinetic energy i not alway conerved? a. Ue the conervation of oentu to calculate the final velocity of the.00 kg object. Before After = / = 1.50 / = +.10 / 1 =.00 kg = 3.00 kg 1 =.00 kg = 3.00 kg v v v v 1v 1 v v v 1.00kg( 4.00 ) 3.00kg( 1.50 ) 3.00kg(.10 ) v 1.00kg v 1.40 or 1.40 wet b. To deterine if the colliion wa elatic, calculate the kinetic energy before the colliion and after the colliion and coare the value. E E E ki ki1 ki E v v 1 1 ki 1 i1 i E (.00 )(4.00 ) (3.00 )( 1.50 ) E 1 1 ki kg kg ki J E E E kf kf1 kf E v v 1 1 kf 1 f1 f E (.00 )( 1.40 ) (3.00 )(.10 ) 1 1 kf kg kg Since the final kinetic energy i le than the initial kinetic energy, the colliion i inelatic. c. Moentu in an iolated yte i alway conerved. Thi law i a direct reult of Newton third law of otion where the change in oentu of one object i equal and ooite the change in oentu of the other object. The cobined change in oentu i zero. Kinetic energy, on the other hand, can eaily be tranfored into heat, ound, and the deforation of object. The Conervation of Energy (i.e. the total aount of energy in an iolated yte doe not change, but energy can be tranfored into other tye) doe not deend on Newton Law. E kf J Dr. Ron Licht 1 7

8 VI. Exloion The riary difference between colliion and exloion i that rior to an exloion there i one object, and after the exloion there are everal object. Exale 7 A 5.0 kg rifle fire a 0.00 kg bullet to the eat. If the uzzle eed of the bullet i 400 /, what i the recoil velocity of the rifle? Note that initially the bullet and rifle are at ret. Therefore = v v v v kg( 400 ) v 1 5.0kg v 1.6 wet Exale 8 A 4.00 kg bob i rolling acro a floor at 3.00 / to the eat. After the bob exlode a 1.50 kg iece ha a velocity of 15.0 / eat. What i the velocity of the other iece? v v v v v v v kg( 3.00 ) 1.50kg( 15.0 ).00kg v 4.0 wet Dr. Ron Licht 1 8

9 Exale 9 In the diagra below, ball A (a = 5.0 kg) i traveling to the left at 1 / and collide with a tationary ball B (a = 4.0 kg). After the colliion ball A i oving at 0.50 / to the left. How high u the incline will ball B rie before coing to ret? d 0 o B A The firt art of the roble involve a colliion between ball A and ball B. Thu we ue the conervation of oentu to find the final eed of ball B. Once we know the eed of ball B, we can ue the conervation of energy to deterine how high it rie on the incline. Part A Find v B B ' ' A A B B A A A ' ' A A A A B B ' B ' B v v v v v v v 5.0kg( 1 ) 5.0kg( 0.50 ) 4.0kg v left Part B Find the height u the ra The kinetic energy of ball B after the colliion i converted into otential energy (h). E 1 k h h E v 1 v g gh ( ) h 10.5 h in0 d h d in d in0 d 31 Dr. Ron Licht 1 9

10 VII. Practice Proble 1. A 0.5 kg ball travelling eat at 4.5 / collide with a 0.30 kg teel ball travelling wet at 5.0 /. After the colliion the 0.30 kg ball i travelling eat with a eed of 0.40 /. What i the final velocity of the 0.5 kg ball? (1.98 / wet). A freight train i being aebled in a witching yard. Boxcar #1 ha a a of 6.4 x 10 4 kg and ove with a velocity of /. Boxcar #, with a a of 9. x 10 4 kg and a velocity of +1. /, overtake Boxcar #1 and coule with it. Neglecting friction, find the coon velocity of the two car after they have couled. (+1.0 /) Dr. Ron Licht

11 3. Starting fro ret, two kater uh off againt each other on ooth, level ice. One i a woan ( = 54 kg) and the other i a an ( = 88 kg). After uhing off, the woan ove away with a velocity of +.5 /. Find the recoil velocity of the an. ( 1.5 /) VIII. Hand-in Aignent 1. Exlain, in your own word, the difference between oentu and inertia.. What i the oentu of a 6.0 kg bowling ball with a velocity of. / [S]? (13. kg / [S]) 3. The oentu of a 75 g bullet i 9.00 kg / [N]. What i the velocity of the bullet? (10 / [N]) 4. A hockey uck ha a oentu of 3.8 kg / [E]. If it eed i 4 / what i the a of the uck? (1.6 x 10-1 kg) 5. A jet flie wet at 190 /. (a) What i the oentu of the jet if it total a i 50 kg? (4.8 x10 5 kg / [W]) (b) What would be the oentu of the jet if the a wa 4 tie it original value and the eed increaed to 6 tie it original value? (1.03 x10 7 kg / [W]) 6. A 30.0 kg object oving to the right at a velocity of.00 / collide with a 0.0 kg object oving to the left with a eed of 6.00 /. If the 0.0 kg object rebound to the right with a eed of 0.75 /, what i the final velocity of the 30.0 kg object? (.50 / [left]) 7. A 5 g ball with a velocity of 40.0 c/ [right] collide with a 15 g ball oving with a velocity of 15.0 c/ [right]. After the colliion the velocity of the 15 g ball wa 35.0 c/ [right]. What wa the velocity of the 5 g ball after the colliion? Wa the colliion elatic or inelatic? (8.9 c/ [right], inelatic) Dr. Ron Licht

12 8. A 95 kg car oving with a velocity of +0.0 / collide with a tationary truck of unknown a. The vehicle lock together and ove off with a velocity of /. What wa the a of the truck? (1.8 x 10 3 kg) 9. In a football gae, a receiver catche a ball while he i tanding till. Before he can ove, a tackler, running at 4.75 /, grab hi. The tackler hold onto the receiver and the two ove off with a eed of.50 /. If the a of the tackler i 15 kg, what i the a of the receiver? (113 kg) 10. An arrow travelling at 45 / trike and ebed itelf in a 450 g ale which i initially at ret. They ove off horizontally at 1 / after iact. What i the a of the arrow? (164 g) 11. A 1.0 x 10 5 N truck oving at a velocity of 17 / north collide head on with a 1.0 x 10 4 N car oving at 9 / outh. If they tick together, what i the final velocity of the car? (1.8 / north) 1. An atronaut i otionle in outer ace. Uon coand, the roulion unit traed to her back eject ga with a velocity of +16 / cauing the atronaut to recoil with a velocity of 0.55 /. After the ga i ejected, the atronaut ha a a of 160 kg. What i the a of the ejected ga? (5.5 kg) 13. A two-tage rocket ove in ace at a contant velocity of /. The two tage are then earated by a all exloive charge laced between the. Iediately after the exloion the velocity of the 100 kg uer tage i /. What i the velocity of the 400 kg lower tage after the exloion? (+4350 /) 14. In the Jae Bond ovie Diaond are Forever the lead feale character fire a achine gun while tanding on the edge of an off-hore drilling rig. A he fire the gun he i driven back over the edge into the ea. The a of a bullet i 10 g and it velocity i 750 /. If her a (including the gun) i 55 kg, what recoil velocity doe he acquire in reone to a ingle hot fro a tationary oition? ( 0.14 /) 15. An ato of uraniu diintegrate into two article. One article ha a a 60 tie a great a the other. If the larger article ove to the left with a eed of.3 x 10 4 /, with what velocity doe the lighter article ove? (+1.4 x 10 6 /) 16. A ballitic endulu i a laboratory device which ight be ued to calculate the velocity of a bullet. A 5.0 g bullet i caught by a 500 g uended a. After iact, the two ae ove together until they to at a oint 65 above the oint of iact. What wa the velocity of the bullet rior to iact? How uch of the original kinetic energy of the bullet energy wa converted into heat? (114 /, -3. J) V =? 65 Dr. Ron Licht 1 1

13 *17. A 70 kg boy it in a 30 kg canoe at ret on the water. He hold two cannon ball, each of a 10 kg. He ick the u and throw both together to the tern of the canoe. The two ball leave hi hand with a velocity of 5.0 / relative to the canoe. A 50 kg girl it in a 50 kg canoe alo at ret on the water. She alo hold two 10 kg cannon ball. However he throw the over the tern of her canoe one at a tie, each ball leaving her hand with a velocity of 5.0 / relative to the canoe. Auing negligible friction between the water and the canoe (a oor aution), calculate the final velocity for each canoe. (Hint: Decribe the velocitie relative to the water, not the canoe.) (0.83 /, 0.87 /) *18. Two en, each with a a of 100 kg, tand on a cart of a 300 kg. The cart can roll with negligible friction along a north-outh track and everything i initially at ret. One an run toward the north and ju off the cart with a eed of 5.00 / relative to the cart. After he ha jued, the econd an run outh and ju off the cart again with a eed of 5.0 / relative to the cart. Calculate the eed and direction of the cart after both en have jued off. (0.5 / North) *19. An argon ato with a a of x 10-6 kg travel at / [right] and elatically trike an oxygen ato with a a of.67 x 10-6 kg dead centre travelling at 0.00 / [left]. What are the final velocitie of each ato? (4.143 / [left], 3.86 / [right]) Dr. Ron Licht

14 Getting tarted Moentu in One Dienion Activity Thi activity involve the ue of an alet. Google het fro the Univerity of Colorado. Find the Colliion alet and download/run it. Ue the Introduction alet. We will ue the Advanced one in the next leon. Play with the alet for a while. Learn how to anually et the a and velocity of each uck and to et the elaticity (e) value. When e = 1 both oentu and kinetic energy are conerved. When e = 0 the uck tick together. When e ha a value between 0 and 1, oentu i conerved but kinetic energy i not conerved. Conervation of Moentu Any object that i oving ha velocity, and alo oentu. But what haen if two or ore object collide? What haen to the velocity or the oentu of each object? Let' exlore thi quetion with the alet. Perfor three different colliion For each colliion, ake ure that each uck ha a different a. One colliion hould involve the red uck coing to a to or continuing to ove in it original direction. One colliion hould involve the red uck rebounding back in the ooite direction. One colliion hould involve the uck ticking together. and colete the following table: Dr. Ron Licht

15 Colliion 1 Object a (kg) v initial (/) v final (/) v (/) initial (kg /) final (kg /) (kg /) Red Green Colliion Object a (kg) v initial (/) v final (/) v (/) initial (kg /) final (kg /) (kg /) Red Green Colliion 3 Object a (kg) v initial (/) v final (/) v (/) initial (kg /) final (kg /) (kg /) Red Green Now, look at the inforation you recorded and calculated in the reviou quetion. What relationhi exit between what haen to the red a and the green a? a. I there a relationhi between the change in velocity of the red a and the green a? If ye, decribe the relationhi. b. I there a relationhi between the change in oentu of the red a and the green a? If ye, decribe the relationhi. c. You hould ee an intereting connection between the change in oentu of each a. If the change in oentu of one object i exactly equal, but ooite to the change in oentu of another object, what doe that indicate about the total oentu of the yte? Dr. Ron Licht

16 If the total oentu of a yte before a colliion i equal to the total oentu of a yte after a colliion, then we ay that oentu i conerved. For the three colliion verify that the total initial oentu i equal to the total final oentu: Note: initial final initialgreenuck finalgreenuck initialblue uck final blue uck Colliion # total initial oentu initial (kg /) total final oentu final (kg /) I oentu conerved? initial final 1 3 Dr. Ron Licht