Discover the answer to this question in this chapter.

Transcription

1 At liftoff, a rocket initially at ret tart to eject ga with a peed of v ep = 000 /. The ga i ejected at a rate of 1000 kg/. The initial a of the rocket i 100 ton, including 60 ton of ga to be ejected. What i the peed of the rocket 30 econd after liftoff? rootfun.net/naphot/apollo-15-iion-gallery-iage/attachent/iile-of-apollo-15-clearing-the-tower/ Dicover the anwer to thi quetion in thi chapter.

2 Luc Treblay Ipule given by a contant force Firt, the definition of ipule will be given. The uefulne of thi quantity will be hown later. If a contant force i eerted on an object during tie t, then the ipule given to the object i Ipule Given to an Object ( ) I = F t The unit for ipule i N or kg /. No other nae wa given to thi group of unit. Mot of the tie, it i ore convenient to work with the coponent of thi equation. Coponent of the Ipule Given to an Object I = F t I y = Fy t I z = Fz t If everal force are acting on an object, the u of ipule given by each of the force i the net ipule. In coponent, thi i I = I = I + I + I + net Net Ipule Given to an Object ( net ) 1 3 I = I = I + I + I + net 1 3 I = I = I + I + I + y net y y1 y y3 I = I = I + I + I + z net z z1 z z3 018 Verion 10 - Moentu

3 Luc Treblay Eaple What are the and y-coponent of the net ipule given to thi bo in 3 econd? child-want-puh-wheeled-cart-directionarked-figure-- adultpuh-horizontal-force--q The coponent of the ipule given by each force will be calculated. The coponent of the ipule given by the 100 N force are ( ) ( ) I = F t = 100N co60 3 = 150 kg 1 1 I = F t = 100N in 60 3 = 59.8 kg 1y 1y The coponent of the ipule given by the 140 N force are Therefore, the net ipule i ( ) ( ) I = F t = 140N co 30 3 = kg I = F t = 140N in 30 3 = 10 kg y y I I = = kg kg kg net = = 49.8 kg kg kg y net Ipule given by a variable force If the agnitude of the force change, the calculation of the ipule ut be divided into part where the force i contant. The ipule given in each part are then added together. In coponent, thi u i Ipule Given by a Variable Force Acting on an Object I = F t I = F t I = F t y y z z Contant F Contant F Contant F 018 Verion 10 - Moentu 3

4 Luc Treblay Eaple A force directed toward the right act on an object. The force i 5 N for 5 econd and then 3 N for 1 econd. What i the ipule given to the object? A the force change, the calculation ut be divided into part. The ipule given during the firt part i I1 = F1 t = 5N 5 = 5 kg The ipule given during the econd part i I = F t = 3N 1 = 3 kg Therefore, the total ipule given to the object i 5 kg/ + 3 kg/ = 8 kg/. If the force i contantly changing, can the calculation till be divided into part where the force i contant? Actually, it can. The tie interval ut be divided into very hort tie interval, o hort that they becoe infiniteially all. Then, the coponent of the ipule given during thee hort tie alo becoe infiniteially all and are di = Fdt di y = Fydt di z = Fzdt If all thee infiniteially all ipule are then ued (with an integral), the total ipule given i obtained. Coponent of the Ipule Given by a Variable Force on an Object (Mot General Forula) t t t I = F dt I = F dt I = F dt y y z z t t t Eaple The force N F = 3 t + N act on an object. What i the -coponent of the ipule given to the object between t = 1 and t = 3? 018 Verion 10 - Moentu 4

5 Luc Treblay The -coponent of the ipule i 3 1 N ( 3 ) I = t + N dt N 3 t = + N t ( ) 3 1 ( ) N 3 3 N 3 1 = + N ( 3) + N ( 1) = 16 kg Therefore, the ipule given i 16 kg/. Graphical repreentation of ipule For a one-dienional otion, the definition of ipule with a contant force I = F t indicate that the ipule i the area under the curve of F on a force-veru-tie graph ince the height of the rectangle i F and it width i t. Even if the force i not contant, thi idea reain valid ince the ipule i the integral I t = t Fdt and thi integral i the area under the curve of F on a force-veru-tie graph. 018 Verion 10 - Moentu 5

6 Luc Treblay The ipule given to an object i the area under the curve of the force acting on the object a a function of tie Reeber that the area i negative if it i below the tie ai. The work wa alo equal to the area under the curve of the force, but there i a crucial ditinction between work and ipule. - The work i the area under the curve of F on a force-veru-poition graph. - The ipule i the area under the curve of F on a force-veru-tie graph Verion 10 - Moentu 6

7 Luc Treblay Proof of the Theore Let find out now why it can be ueful to calculate the ipule given to an object. Thi all tart with the definition of ipule. t t t I = F dt I = F dt I = F dt net net y net y net z net z net t t t We will work only on the -coponent. The reult are iilar for the other coponent. Uing the fact that F net = a and that acceleration i the derivative of velocity, the ipule i net t I = F dt = = = = t t t t t v v net a dt dv dt dt dv [ v ] v v = v v The nae oentu wa given to thi new quantity. It i denoted p. p = v By including the other coponent, oentu i Moentu ( ) p = v in coponent, thi i p = v p = v p = v y y z z 018 Verion 10 - Moentu 7

8 Luc Treblay The unit of oentu i alo the kg /. (The agnitude of thi vector v i ued in phyic ince a long tie ago. The ipetu of edieval theorie wa often defined a weight ultiplied by peed. Obviouly, it role changed a lot with Newtonian phyic.) The oentu coponent are calculated with the velocity coponent. In two dienion, thi ean that p = v = v coθ p = v = v inθ y y Coon Mitake: Wrong Sign for p Moentu i a vector. Therefore, it direction i iportant. Be ure to define clearly poitive direction with ae. If the coponent of the oentu i in the direction of your ai, it i poitive, and if it i in the oppoite direction to your ai, it i negative. With thi definition of oentu, the oentu equation becoe I = v v net = p p With the other coponent giving iilar reult, the following theore i thu obtained. Ipule-Moentu Theore I = p net in coponent, thi i I = p I = p I = p net y net y z net z Here an eaple of the application of thi theore. 018 Verion 10 - Moentu 8

9 Luc Treblay Eaple A 5 kg bo lide on a horizontal urface with an initial velocity of 0 /. The coefficient of friction between the urface and the bo i 0.1. What i the peed of the bo 5 econd later? I net Calculation In order to find the ipule given to the bo, the force acting on the bo ut be found. The force are: 1) The weight (49 N) directed downward. ) The noral force (49 N) directed upward. 3) The friction force ( N = 4.9 N) directed toward the left. The weight and the noral force cancel each other, and the y-coponent of the net force i, therefore, zero. Thi ean that the y-coponent of the ipule given alo vanihe and that the y-coponent of the velocity doe not change. Thu, the y-coponent of the velocity reain zero. The only force with an -coponent i the friction force. The -coponent of the net ipule i, therefore, p Calculation I = F t f f ( N ) = = 4.5 The -coponent of the oentu change i p = v v kg = 5kg v 5kg 0 = 5kg v 100 kg 018 Verion 10 - Moentu 9

10 Luc Treblay Ipule-Moentu Theore Application net kg 4.5 = 5kg v 100 I = p v = 15.1 kg Thi proble-olving ethod i quite correct, but it i not ued very often. Mot of the tie, thi kind of proble i rather olved by finding the acceleration with Newton econd law and then by finding the velocity with the acceleration and the tie. What Method Should Be Ued to Solve a Proble? Norally, you hould have the ipreion that proble-olving with I net p i trangely iilar to proble-olving with W net E k. Perhap you are wondering which ethod to ue. A work i force ultiplied by ditance, it i eaier to olve proble with W net E k if you are aked to find oething after an object ha travelled a certain ditance. A ipule i force ultiplied by tie, it i eaier to olve proble with I net p if you are aked to find oething after a certain tie. Obviouly, you can alo olve any proble by finding the acceleration with Newton econd law or by uing the law of echanical energy conervation. Obviouly, thee 4 ethod all give the ae reult, but oe of the require a lot ore calculation if they are not ued in the right contet. A More General Second Law In a previou chapter, power wa defined a being the work done divided by the tie required to do thi work. Let ee what happen if the ipule i divided by the tie required to give thi ipule to an object. For a contant force, thi i I F t = = F t t The reulting quantity i not new; it i iply the force acting on the object. 018 Verion 10 - Moentu 10

11 Luc Treblay Since Inet = p, thi alo ean that F net Inet = t i Relationhip between Force and Moentu (Contant Force) F net p = t If the force i not contant, the ipule can be calculated for an infiniteially all tie dt. During uch a hort tie, the variation of oentu i alo infiniteially all and i dp. The end reult i a ore general verion of Newton econd law. Newton Second Law (More General) F net dp = dt Leonard Euler propoed thi forula to iprove Newton econd law in 175. It i better than F net = a a it applie to ituation where the a of the object varie, like a rocket, for eaple. It i equivalent to F net = a only if the force act on a contant a object. In uch a cae, the forula give That wa the econd law o far. dp Fnet = dt d v Fnet = dt dv Fnet = dt F = a net ( ) 018 Verion 10 - Moentu 11

12 Luc Treblay Graphical interpretation If force i the derivative of oentu, then the force eerted on the object i the lope on a graph of oentu-veru-tie. Average force The average force i defined a being a contant force that give the ae ipule to an object during the ae tie the force ha acted. Thi ean that the area under the curve ut be the ae for thee two graph. (By the way, thi i how the average value of any function on an interval i defined in atheatic.) A the area i equal to the ipule, thi ean that I = F t The ae reult are obtained for the other coponent. Therefore, Average Force Acting on an Object I p I p I p F = = F = = F = = t t t t t t y y z z y z 018 Verion 10 - Moentu 1

13 Luc Treblay Eaple A 150 g baeball oving toward the left at 45 / i hit by a bat. After the colliion, the ball goe at 60 / toward the right. What i the average force eerted on the ball if the ipact between the ball and the bat lat 0.01? The average force i calculated with the following forula. F I = t The ipule given to the ball ut then be found firt to calculate the force. With a poitive -ai pointing toward the right, the ipule given to the ball i I = p = p p ( ) = 0.15kg kg 45 = kg Forgetting to conider the ign of the velocity i a relatively coon itake in thi kind of calculation. Therefore, the average force i F kg = I N t = 0.01 = The average force i then 1575 N toward the right (ince the reult i poitive). Eaple Bullet are hot at a wall with a achine gun. 600 bullet are hot per inute. Each ball ha a a of 10 g and a peed of 800 /. What i the average force that the bullet eert on the wall? Verion 10 - Moentu 13

14 Luc Treblay The average force ade by the wall on the bullet (required to top the) will be calculated firt. To do thi, the ipule given to the bullet in a inute will be calculated. A poitive ai pointing to the right i ued. F on bullet p = t p p = t kg 800 = 60 = 80N The final oentu i zero ince the ball are now at ret in the wall. A force of 80 N directed toward the left i thu obtained. Reeber that thi i an average force. The force i zero when no bullet are hitting the wall and rie uddenly when a bullet hit. The average of thi changing force i 80 N. According to Newton third law, the bullet eert an average force of 80 N toward the right on the wall if the wall eert an average force of 80 N toward the left on bullet. The anwer i, therefore: 80 N toward the right. Eaple A car going at 7 k/h hit a wall. During the accident, the paenger top over a ditance of 1 (becaue the front of the car cruple 1 ). What i the average force eerted on a 60 kg paenger during the colliion? The duration of the colliion ut firt be calculated uing kineatic forula. The average force can then be calculated. 1 = 0 + ( v0 + v) t 1 1 = 0 + ( 0 + 0) t t = 0.1 p F = t p p = t 0 60kg 0 = = 1,000N Verion 10 - Moentu 14

15 Luc Treblay The force i negative becaue an ai in the direction of the velocity wa ued. A negative force ean that the force i in the oppoite direction to the velocity, o toward the rear of the car. Note that thi force correpond to nearly 0 tie the weight of the peron. Thi peron thu eperience around 0 g during the colliion. It i poible to urvive thi becaue a huan being can urvive a crah if he eperience le than about 100 g. If the paenger i not wearing a eatbelt, the reult change a lot, not becaue the change of oentu increae but becaue t will be uch aller. Without a eatbelt, the car tart to low down wherea the peron continue to ove according to Newton firt law. He top only when he coe into contact with the teering wheel or the windhield. The topping ditance i then about one centietre, and the topping tie i about Thi ean that the average force i then about 1. illion N, or about 000 tie the weight of the peron. No one can urvive uch large force acting on hi body. The airbag doe the ae thing a the eatbelt; it prevent you fro continuing your otion and crah on the dahboard. It force you to low down at the ae rate a the car in order to reduce the average force. Alo, it would be ridiculou to have a car o tiff that it doe not cruple on ipact. If a car were o rigid o that it would cruple only a centietre when it hit a wall, it would aount to the ituation where the topping tie wa only 0.001, and the force would be 1. illion N, even if the paenger are wearing their eatbelt. Note that the previou eaple ight have been olved with the work forula. A the average could alo have been defined with the following forula W = E = F coθ the calculation of the average force would have been k F ( ) E k v f vi 0 60kg 0 = = = = 1,000N coθ coθ 1 co180 Here, a poitive repone i obtained ince the agnitude of the force i found with thi forula. However, the direction of the force wa taken into account ince 180 a ued for the angle between the otion and the force. It wa thu aued that the force wa in the oppoite direction to the otion. 018 Verion 10 - Moentu 15

16 Luc Treblay Proof of the Conervation Principle The ipule-oentu theore i Since the net ipule i the theore becoe p = I net I = F t net net p = F t net Thu, if the net force i zero, the following reult are obtained. p = 0 p = contant Thi reult i not really urpriing ince it wa already known by Newton firt law that the velocity i contant if the u of the force acting on an object i zero. If the velocity i contant, then that the oentu i contant. However, an intereting reult i obtained if the ae tep are ade by uing the ipule eerted on all the object of a yte. Thi u i yte yte net I net = p yte yte I = p p ( ) F t = p p net yte yte yte To the right, thee are the u of the oentu of all the object in the yte. Thi i the total oentu of the yte that will be denoted p tot. The equation i then F t = p p yte net tot tot A t i the ae for all the force, it can be taken outide the u. 018 Verion 10 - Moentu 16

17 Luc Treblay F t = p p yte net tot tot The u of the net force acting on the yte i iply the u of all the force acting on the yte. Therefore, the equation can be written a F t = p tot p yte The force acting on an object of the yte can be internal (ade by another object in the yte) or eternal (ade by an object fro outide the yte). tot F t F + t = p p et int tot tot yte yte However, if object A eert a force on object B, then object B eert a force of the ae agnitude and oppoite direction on object A according to Newton third law. Thi ean that when the u of the internal force acting on all object in the yte i done, thee two force alway cancel each other becaue the two object are part of the yte. It alo ean that the u of eternal force i not necearily zero ince only one of the object i part of the yte and that the two force aociated by Newton third law cannot cancel each other. The equation thu becoe F t = p p yte et tot tot If the u of the eternal force i zero, then = p p 0 tot tot Thi ean that the total oentu of the yte reain contant then. Therefore, Moentu Conervation Law p tot = contant if F et = 0 Reeber, however, that thi law of conervation i not alway true: It i only true if the u of all eternal force i zero. There ay be eternal force, but the u of thee force ut be zero for the law of oentu conervation to hold. 018 Verion 10 - Moentu 17

18 Luc Treblay In coponent, thi law i Moentu conervation law p = p if F = 0 tot tot et yte p = p if F = 0 y tot y tot y et yte p = p if F = 0 z tot z tot z et yte Note that thee three equation are independent of each other. Thi ean that it i poible for the -coponent of oentu to be conerved while the y-coponent i not conerved. Application of the Law Thi law of conervation can now be ued for proble-olving. Certain type of proble can be iply olved with thi law, including colliion proble. Reolution Method 1) The coponent of the total oentu of the yte are calculated at oe intant. Thee coponent are noted p and p y. In one dienion, there i only p. In two dienion, both p and p y are ued. There ight be a third dienion, but we will not go o far in thee note. ) The coponent of the total oentu of the yte are calculated at another intant. Thee coponent are noted p' and p' y. 3) The conervation law i then applied p p y = = p p y 4) The equation are olved. Reeber that in order to apply correctly the law of conervation of oentu in thi for, the net eternal force ut be zero. There ay be eternal force, but their u ut vanih. 018 Verion 10 - Moentu 18

19 Luc Treblay Eaple An 80 kg atronaut in pace hold of 4 kg bowling ball in hi hand. He i initially otionle in pace. The atronaut then throw the bowling ball o that the ball now ove with a velocity of 5 /. What i the velocity of the atronaut after he had thrown the ball? Here the ituation at intant 1 and. Originally (intant 1), the -coponent in the total oentu of the yte (peron and ball) i p = = 0 ince the atronaut and the ball are both at ret. At intant, the -coponent in the total oentu of the yte i p = v + v = 80kg v + 4kg 5 A A B B A The oentu conervation law can now be applied a there i no eternal force. The only force here are the force between the ball and the atronaut. Since both thee object are part of the yte, thee force are internal force. The conervation law then give 0 = 80kg v + 4kg 5 A v = 0.5 A p = p Thu, the atronaut ove at 0.5 / toward the left. It i not o urpriing to have an atronaut oving toward the left according to Newton' third law. When the atronaut puhe the ball toward the right to give it peed, the ball puhe on the atronaut toward the left with a force of the ae agnitude. Thi force then give a peed the atronaut toward the left. 018 Verion 10 - Moentu 19

20 Luc Treblay Thi i baically how a rocket i propelled. The atronaut peed increae when he throw a ball. Every tie he throw a ball (alway in the ae direction), it peed increae. Here a bad pacehip odel, propelled by bowling ball. The atronaut in the rocket iply throw bowling ball to give peed to the pacecraft! Whenever the atronaut launche a bowling ball, the peed of the hip increae a little. The ore bowling ball he throw and the fater he throw the, the fater the pacehip will ove at the end. Thi i not eactly how it i done but it i not that far fro the truth. Intead of throwing bowling ball, ga olecule are ejected. Due to it low a, one thrown olecule doe not increae the peed of the hip uch. However, if enough of the are ejected, the pacehip could end up with a large peed. Thi i how Dr. Sith propelled hielf in thi video. Thi video i aid to be a deontration of Newton third law, but thi i alo a deontration of the conervation of oentu. Thi i alo what happen when you let go an inflated balloon. When the air i ejected in one direction, the balloon i propelled in the oppoite direction. Thi idea can be ued to propel a all vehicle. A larger peed can be achieved if the ga olecule are ejected with ore peed, which can be done by heating the ga. An eotheral cheical reaction between ubtance can for a very hot ga. Ejected in one direction, thi ga propel the vehicle in the oppoite direction. Thi i what can be een in thi video or thi one We will return later to thi ubject and calculate the peed of a rocket ejecting ga. 018 Verion 10 - Moentu 0

21 Luc Treblay Eaple A 60 kg peron at ret on a kateboard catche a baeball ( = kg) going at 160 k/h. What will be the peed of the peron (with the ball in hi hand) after the catch? At intant 1, the oentu of the yte (peron and ball) i (uing the ai hown in the figure) p = v + v ball ball peron peron = 0.135kg = 6 kg At intant, the oentu of the yte i p = v + v ball ball peron peron = 0.135kg v + 60kg v = ( ) kg v The velocity can be found with the law of oentu conervation. ( ) kg 6 = 60,135kg v p = p v = 0,09976 In thi eaple, it i iportant to note that there are eternal force: the weight and noral force. A the yte i fored of the ball and the peron, the force ade by the Earth and the ground are eternal force. However, the conervation law can till be applied, ince thee two force cancel each other, and the u of the eternal force i zero. It can alo be een that by receiving an object, a peron gain peed. Thi how that a peron can change hi peed not only by throwing oething but alo by catching oething. By cobining thee two ethod, the peron in thi video can ove Verion 10 - Moentu 1

22 Luc Treblay She i on a kind of air cuhion in order to eliinate the friction force which i an eternal force. Gun Recoil If there i an eploion, the force ade by the ga releaed by the eploion i an internal force between object in the yte if the ga i included in the yte. Take the eaple of a cannon hooting a hell. Initially, the gun, the hell, and the eploive charge are at ret, and the total oentu i zero. When the load eplode, and the hell i propelled, the total oentu of thi yte ut reain zero. To iplify, the ae of the eploive charge and of the releaed ga are neglected o that only the oentu of the gun and the hell are conidered. If the ai i in the direction of otion of the hell, then the oentu of the hell i poitive. The oentu of the cannon ut then be negative (for the u to be zero), which ean that the gun ove in a direction oppoite to the hell velocity. Thi otion i the recoil of the gun. Eaple A 500 kg cannon, initially at ret, fire a 5 kg hell with a peed of 500 /. What i the velocity of the canon after the firing of the hell? At intant 1, the oentu i (uing a poitive ai pointing toward the right) p = v + v cannon cannon hell hell = = 0 kg At intant, the total oentu i p = v + v cannon cannon hell hell = 500kg v + 5kg 500 cannon ction3.rhtl Uing the law of oentu conervation, the velocity i The gun then recoil at 5 /. p 0 = 500kg v + 5kg 500 cannon v = 5 cannon = p 018 Verion 10 - Moentu

23 Luc Treblay An eaple of gun recoil can be een in thi video. Do not reain behind a gun when it i fired. Otherwie, here what could happen to you. Gun and rifle alo recoil when fired. The larger the oentu of the bullet, the larger the recoil i becaue both oentu ut cancel each other ince the initial oentu wa zero. With a gun that give a large oentu to the bullet, the recoil can be difficult to control. Movie Mitake When people get hot in a ovie, they are often thrown back over a ignificant ditance after ipact. Thi i the cae in thi ecerpt fro the fil Martyr, a French horror fil. No need to watch the whole clip, the firt 0 econd i ufficient. It i poible for a huan being to be thrown back a far a i the father of thi faily, but the bullet ut then have a gigantic oentu. A the 70 kg father (approiately) i being thrown back at 5 / (approiately), he receive a oentu roughly equal to 70 kg 5 / = 350 kg/. Thi i really a coloal oentu ince it i equivalent to a 100 g projectile going ten tie fater than ound! Even o, let aue thi ight be poible. Before the hot i fired, the oentu i zero (hooter, rifle, and bullet at ret). If the projectile i fired with a oentu of 350 kg/, the rifle and the hooter ut recoil with a oentu of 350 kg/ in the oppoite direction o that the total oentu of the yte reain zero. If the hooter and rifle received 350 kg/ of oentu, then they would be thrown back with great peed. You urely notice that it i the ae oentu a the one received by the victi. Thu, the hooter hould be thrown backward a violently a the one who received the hot. Thi i clearly not what happened in the clip. Thi i what really happen when oeone fire a gun giving a coniderable oentu to a bullet. Thi gun give a oentu of 37 kg/ to the bullet. Iagine if it had given 350 kg/! 018 Verion 10 - Moentu 3

24 Luc Treblay Other Eaple of the Application of the Law of Moentu Conervation Eaple A 10 kg dog i on a 30 kg raft. Initially, the raft and dog are otionle. Then the dog tart walking toward the left with a peed of 6 /. What i the velocity of the raft? aaugh.co/wordpre/010/1/oentou-peanut/ Initially, the total oentu of the yte (dog and raft) i (uing a poitive ai toward the right) p = v + v raft raft dog dog = = 0 kg When the dog i walking toward the left, the total oentu of the yte i p = v + v raft raft dog dog ( ) = 30kg v + 10kg 6 The law of oentu conervation then give p raft ( ) 0 = 30kg v + 10kg 6 raft v = raft = p The raft i thu oving at / toward the right. 018 Verion 10 - Moentu 4

25 Luc Treblay Eaple A 100 kg bob oving at 5 / eplode into three fragent. If the peed and the direction of the velocitie of the 30 kg and 45 kg fragent are thoe indicated in the figure, what i the velocity of the 5 kg fragent? Let' tart with the -coponent of the oentu. Initially, the -coponent of the total oentu i p = v = 100kg 5 = kg After the eploion, the -coponent of the total oentu i p = v + v + v ( ) = 45kg kg 5 co kg v The law of oentu conervation then give 3 3 ( ) 3 kg 500 = 45kg kg 5 co kg v p = p v = 1.79 Initially, the y-coponent of the total oentu i 018 Verion 10 - Moentu 5

26 Luc Treblay p y = v y = 100kg 0 = 0 After the eploion, the y-coponent of the total oentu i p = v + v + v y 1 1y y 3 3y ( ) = 45kg kg 5 in kg v The law of oentu conervation then give 3y y ( ) 3 0 = 45kg kg 5 in kg v p = p y v = 1.1 Fro the coponent, the peed can be found v = v + v y ( 1.79 ) ( 1.1 ) = + = 4.77 and the direction of the velocity can be found v θ = arctan v 3y = arctan 1.79 = 11.1 (180 ha been added to the anwer given by the calculator becaue v' 3 i negative.) 3y y 018 Verion 10 - Moentu 6

27 Luc Treblay During a colliion, there are force between the two object in contact. If the yte conit of the two object colliding, thee force are internal force and the total oentu ut be conerved in the colliion. Therefore, Moentu and Colliion In a colliion, the total oentu of the colliding object i conerved. Therefore, the oentu of the yte before the colliion i the ae a the oentu after the colliion. p = p Thi law i oetie applied to a colliion even if there i an eternal force. Thi can be done becaue the latter i often negligible copared to the force between object during the colliion. Thu, when a baeball player trike a ball, the law of conervation of oentu i applied depite the preence of the force of gravity on the bat and on the ball (which are eternal force). The force eerted on the ball and the bat are o uch larger than the force of gravity that the effect of the gravitational force during the colliion can be neglected. However, the law of oentu conervation i not ufficient to fully reolve colliion proble, oe other inforation ut be provided. It i neceary to know how the two object behave when they coe into contact. Both object can tick together or can bounce off each other upon contact. Copletely Inelatic Colliion In a copletely inelatic colliion, the two object tick together after the colliion and, therefore, have the ae velocity. In uch a cae, the law of oentu conervation i ufficient to olve the proble. Copletely Inelatic Colliion In a copletely inelatic colliion, the two object tick together and have the ae velocity after the colliion. Only the oentu of the yte i conerved then. p = p 018 Verion 10 - Moentu 7

28 Luc Treblay Eaple The two vehicle hown in the figure are involved in a colliion. What will be the velocity of the vehicle after the colliion if they tick together? Before the colliion, the -coponent of the total oentu i p = v + v 1 1 fr.depoitphoto.co/577683/tock-illutration-car.htl ( ) = 100kg kg = kg / After the colliion, we have a ingle object of 6600 kg, the -coponent of it oentu i p = v = 6600kg v The law of oentu conervation then give p = p kg / = 6600kg v v = / Ballitic Pendulu A ballitic pendulu i ued to eaure the peed of certain object, like rifle bullet, for eaple. When the object collide with the pendulu initially at ret, it give a certain peed to the pendulu which then wing up to a aiu angle. Fro the aiu angle, the peed of the object before the colliion with the pendulu can be calculated. Here i a video howing a ballitic pendulu in action Verion 10 - Moentu 8

29 Luc Treblay Eaple A bullet ( 1 = 35 g) i fired into a wooden block ( = kg) which i hanging at the end of a 160 c long rope. The pendulu then wing up to an angle of 40. What wa the peed of the bullet? (There will be no prie for the quantitie ued in configuration a, one prie for the quantitie ued in configuration b and two prie for the quantitie ued in the configuration c.) Before applying the law of oentu conervation for the inelatic colliion between the bullet and the wooden block, the peed of the pendulu iediately after the colliion ut be found knowing that it wing up to 40. Thi peed i found with the law of echanical energy conervation. Iediately after the colliion (figure b), the echanical energy i (placing the y = 0 at the lowet point of the pendulu) 1 = + 1 E v gy = v ' When the angle i aiu (figure c), the echanical energy i 1 = gy E = v + gy a The law of echanical energy conervation then give 018 Verion 10 - Moentu 9

30 Luc Treblay 1 v ' E = E = gy v ' = gy a a The aiu height reached by the pendulu can be calculated fro the aiu angle with y a ( 1 coθa ) ( ) = L = co 40 = The peed of the block right after the colliion i then v' = gy a = =.709 ² The colliion between the bullet and the wooden block can now be eained. Before the colliion (figure a), the -coponent of the total oentu of the bullet-block yte i p = v + v bullet bullet block block = 0.035kg v + 0 After the colliion (figure b), we have a ingle object of,035 kg (the block with the ebedded bullet). The -coponent of the total oentu i p = v ' bullet =.035kg.709 = 5.51 The law of oentu conervation then give u the peed of the ball. kg 0.035kg v = 5.51 v p bullet = p bullet = kg 018 Verion 10 - Moentu 30

31 Luc Treblay Elatic Colliion In an elatic colliion, the two colliding object bounce off each other without any lo of kinetic energy. To illutrate, iagine that a ball i dropped and then bounce off the floor. If the colliion i elatic, then the kinetic energy of the ball jut after the colliion i the ae a it wa jut before the colliion. A the kinetic energy before the colliion coe fro gravitational energy and the kinetic energy after the colliion return to gravitational energy, the gravitational energie at the highet point are the ae before and after the colliion, which ean that, after the colliion, the ball will acend to the ae height it wa releaed. Elatic Colliion In an elatic colliion, the oentu and the kinetic energy are the ae before and after the colliion. p E k = p = E k Eaple A 1 kg ball oving at 15 / toward the right collide head-on with a kg ball oving toward the left at 6 /. What are the velocitie of the ball after the colliion if the colliion i elatic? The law of oentu conervation give ( ) 1kg 15 + kg 6 = 1kg v + kg v 3 = v + v p = p v + v = v + v 1 chool.wikia.co/wiki/moentu:_colliion Verion 10 - Moentu 31

32 Luc Treblay The kinetic energy conervation give E = E k v 1 + v = 1v 1 + v ( ) ( ) 97 = v1 + v k 1 1kg 15 + kg 6 = 1kg v + kg v Thee two equation ut be olved in order to find the velocitie. We ll tart by olving for v' 1 in the oentu equation v = 3 v 1 and then ubtitute thi value into the kinetic energy equation. 97 = v1 + v ( v ) 97 = 3 + v 1 97 = 9 1 v + 4v + v 6v 1 v 88 = 0 Thi quadratic equation can then be olved. The olution are v = 8 and v = 6 There are alway two olution, and one of the i alway identical to the velocity before the colliion. The olution of the conervation law correpond to all the velocitie for which the oentu and the kinetic energy are the ae a they were before the colliion. Obviouly, the velocitie before the colliion give the ae oentu and kinetic energy a the one before the colliion! The velocity after the colliion i the olution which i different fro the initial velocity. Therefore, v = 8 Then the velocity of the other ball can be found. The oentu equation i ued, becaue we ll have to gue the ign of the velocity if the kinetic energy equation i ued becaue the velocity i quared. Therefore, the velocity of the other ball i 3 = v + v = v + 8 v = Verion 10 - Moentu 3

33 Luc Treblay The velocitie after the elatic colliion are hown in the following figure. chool.wikia.co/wiki/moentu:_colliion Inelatic Colliion (but not Copletely Inelatic) Mot of the tie, colliion are not elatic even if the object do not tick together. A oe kinetic energy i lot in a colliion (in the for of a peranent deforation or ound or heat, for eaple), the kinetic energy i lower after the colliion. The conervation of kinetic energy thu cannot be ued in thi kind of proble. Thi i an inelatic colliion (ince kinetic energy i not conerved), but it i not copletely inelatic. To olve an inelatic colliion proble, only the equation of oentu conervation can be ued. However, ince there are two velocitie to find (the velocity of each object after the colliion), and there i only one equation, oe additional inforation ut be given. For eaple, thi additional inforation can be the velocity of one of the object after the colliion or the fraction of kinetic energy lot in the colliion. Eaple In an inelatic colliion, a 1 kg ball oving at 5 / toward the right collide head-on with a kg ball oving toward the left at /. After the colliion, the velocity of the 1 kg ball i 4 / toward the left. a) What i the velocity of the kg ball after the colliion? The law of oentu conervation give chool.wikia.co/wiki/moentu:_colliion 018 Verion 10 - Moentu 33

34 Luc Treblay p ( ) ( ) 1kg 5 + kg = 1kg 4 + kg v v =.5 = p v + v = v + v b) What fraction of the kinetic energy i lot in the colliion? The initial total kinetic energy of the ball i 1 1 Ek = 1kg ( 5 ) + kg ( ) = 16.5J The total kinetic energy of the ball after the colliion i 1 1 E k = 1kg ( 4 ) + kg (.5 ) = 14.5J (Of coure, the final kinetic energy after the colliion can never be larger than the kinetic energy before the colliion.) Therefore, the energy lo i The fraction lot i then E = E E = 14.5J 16.5J =.5J k k k E E ki k.5j = = J (It i negative becaue it i a lo.) Therefore, 13.6% of the echanical energy i lot in thi colliion. The condition are the ae in two dienion a they were in one dienion with the eception that there are now two equation for oentu conervation: the conervation of the -coponent of oentu and the conervation of the y-coponent of oentu. For a copletely inelatic colliion, the rule are, therefore, 018 Verion 10 - Moentu 34

35 Luc Treblay Copletely Inelatic Colliion In a copletely inelatic colliion, the two object tick together and have the ae velocity after the colliion. Then, only the oentu of the yte i conerved. p p y = p = p y Eaple A 000 kg car oving north at 30 / collide with a 1500 kg car oving at 0 / in the direction hown in the figure. If both car tick together after the colliion, what i the velocity (agnitude and direction) of the car after the colliion? A thi i a copletely inelatic colliion (ince the two object tick together after the colliion) only the oentu i conerved. Let' tart with the conervation of the -coponent of the oentu. (The -ai i toward the Eat, and the y-ai i toward the North.) 1 1 tot ( ) 000kg kg 0 co 40 = 3500kg v v = p = p v + v = v Then, the conervation of the y-coponent of the oentu give 018 Verion 10 - Moentu 35

36 Luc Treblay Thu, the peed i y 1 1y y tot y ( ) 000kg kg 0 in 40 = 3500kg v y v = y p = p v + v = v v = v + v y ( ) ( ) = + = and the direction of the velocity i v y θ = arctan v = arctan = y For an elatic colliion, the following rule apply. Elatic Colliion In an elatic colliion, the oentu and the kinetic energy are the ae before and after the colliion. p p E y k = p = p y = E k 018 Verion 10 - Moentu 36

37 Luc Treblay Eaple A 100 g ball oving at 6 / in the direction of the -ai collide with an 800 g ball at ret. After the colliion, the 100 g ball i heading in the direction of the y-ai. Deterine the peed of the 100 g ball and the velocity (agnitude and direction) of the 800 g ball if the colliion i elatic. There are now 3 equation: the conervation law for the and y coponent of the oentu and the law of conervation of kinetic energy (ince the colliion i elatic). Three equation are required ince there are three quantitie to find: two peed and the direction of the velocity of the 800 g ball. The equation for the conervation of the -coponent of the oentu i kg = kg v p v = 0.75 = p v + v = v + v The equation for the conervation of the y-coponent of the oentu i p y = p v + v = v + v 1 1y y 1 1y 1 y = 0.1kg v + 0.8kg v y v = 8v 1y y 1y y 018 Verion 10 - Moentu 37

38 Luc Treblay The equation for the conervation of kinetic energy i E = E k v 1 + v = 1v 1 + v ( ) 36 = v1 + 8v k 1 0.1kg = 0.1kg v + 0.8kg v To olve thee equation, it ut be reebered that v = v + v. The energy equation then becoe y = v1 + 8v = ( v 1 + v ) ( 1y + 8 v + v y ) The two reult previouly obtained fro the oentu conervation equation are then ubtituted in thi equation. ( v 1y ) ( v v y ) 36 8 = + + v 1y = ( v 1y ) + ( ) + 8 v 1y 9v 1y 31.5 = v 1y + = 8 8 v = y The lat velocity coponent needed i then found with the y-coponent of the oentu equation. v 1y v y = = Therefore, the agnitude and direction of the 800 g ball velocity are ( ) ( ) y v = v + v = = 1 θ v y = arctan = arctan = v 0.75 The 100 g ball thu ove at 5.9 /, and the 800 g ball ove at 1 / at Verion 10 - Moentu 38

39 Luc Treblay For elatic colliion, one inforation concerning the ball after the colliion ut be given. In the lat eaple, it wa known that the 100 g ball wa oving in the direction of the poitive y-ai. If no additional inforation were given, four unknown would have to be found with three equation (two oentu equation and one kinetic energy equation). Since it i ipoible to find ore unknown than the nuber of equation, it would be ipoible to olve the proble. Doe thi ean that it i ipoible to know the reult of the colliion with only the law of phyic? Of coure not. The additional data given actually infor u about the way the ball hit each other. Depending on the value of b in the figure (called the ipact paraeter) the reult of the colliion i different. Thi value of b ay be the additional data given, and the direction and peed of the two object after the colliion could be calculated uing thi inforation. A thi i quite tricky, the peed or the direction of the velocity of one of the object after the colliion i given in thee note. Thi i equivalent to giving the value of b. worldofoentu.wordpre.co Relationhip between Moentu and Kinetic Energy The kinetic energy can be found directly fro the oentu without finding the peed ince there i a forula to pa directly fro one to the other. Thi forula i obtained in the following way. Thi lead to v Ek = v Ek = E k = ( v) Relationhip between Moentu and Kinetic Energy p Ek = 018 Verion 10 - Moentu 39

40 Luc Treblay Three Difference between Kinetic Energy and Moentu 1) Colliion During a colliion - Moentu i alway conerved. - Kinetic energy i conerved only if the colliion i elatic. ) The Direction of the Velocity The direction of the velocity i iportant, but only for the oentu ince only thi quantity i a vector. Only the peed atter for the kinetic energy, the direction of the velocity ha no ignificance. - Moentu i a vector - Kinetic energy i a calar So it i poible to change the oentu without changing the kinetic energy if the direction of the velocity change while the peed tay the ae. However, it i ipoible to change the kinetic energy without changing the oentu. 3) Rate of Change of Moentu and Kinetic Energy The equation W net = E k indicate that the work correpond to the change in kinetic energy. The rate at which kinetic energy i added i the power. dw P = dt The equation I = p net indicate that the ipule correpond to the change in oentu. The rate at which oentu i added i the force. dp F = dt 018 Verion 10 - Moentu 40

41 Luc Treblay Thi figure uarize thee four forula. Vi Viva Controvery It i quite legitiate to wonder which of thee two quantitie (oentu and kinetic energy) i the ot iportant in phyic. Thi quetion wa at the origin of the vi viva controvery (vi viva wa the nae given to the quantity v². It wa renaed kinetic energy when the ½ wa added in 189.) The controvery began in the 17 th century when the Royal Society launched a copetition to find the law of colliion, a phenoenon of the highet iportance then. At that tie (1666), Newton ha not yet publihed hi law (1686), and it wa hoped that the law of nature were to be dicovered fro colliion. Soe cientit then found that, during a colliion, oentu i conerved when certain condition are et while other found that kinetic energy i conerved when certain condition are et. (Since they were alo trying to define force by tudying colliion, you can now undertand why the force wa often defined a being proportional to v or v² before Newton.) It ay ee urpriing that there wa no agreeent on the conervation of oentu ince we have een that it i alway conerved in a colliion. Thi i becaue oe cientit did not realize that oentu i a vector. They calculated v by iply taking the agnitude of the velocity without conidering it direction. No wonder then that they thought that oentu wan t alway conerved. The abence of conenu concerning the conervation of kinetic energy i le urpriing ince it i only conerved in elatic colliion. Each ide then trie to convince the other that their quantity wa the bet. Many arguent ued in favour of one or the other went beyond iple colliion arguent. For eaple, 018 Verion 10 - Moentu 41

42 Luc Treblay it wa argued that the oentu wa ore fundaental becaue the flight tie of an object i thrown upward double if the oentu double. On the other ide, it wa argued that the kinetic energy wa ore fundaental ince the aiu height of an object thrown upward double i the kinetic energy double. Actually, no quantity i ore fundaental than the other; it depend on what i ought. If tie i an iportant apect of the proble, oentu ee to be ore iportant. For eaple, if car A ha twice the oentu than car B, then car A take twice a uch tie to top then car B. On the other ide, if the ditance i an iportant apect of the proble, the kinetic energy ee to be ore iportant. For eaple, if car A ha twice a uch kinetic energy a car B, then the topping ditance of the car A i twice the topping ditance of car B. The controvery, which lated about 40 year, wa, therefore, quite futile. It i poible to ue oentu to olve a proble even if oentu i not conerved becaue there are eternal force. With a non-vanihing eternal force u, the oentu change i F t = p p yte et tot tot The ter on the left i the net ipule given to the yte by eternal force. The equation i then I = p p et net tot tot Thi i the ipule-oentu theore applied to a yte. In coponent, thi i Ipule-Moentu Theore for a Syte p = p + I tot tot net et p = p + I tot y tot y net et y p = p + I tot z tot z net et z where I net et are net ipule ade by eternal force. 018 Verion 10 - Moentu 4

43 Luc Treblay Eaple A bullet i fired into a wooden block and get lodged in it (which correpond to a copletely inelatic colliion). After the colliion, the block lide on the floor. The kinetic friction coefficient between the block and the floor i 0.3. What i the peed of block econd after being hit by the bullet? If the yte conit of the block and the bullet, the friction force i an eternal force. The proble can then be olved with the forula p tot = ptot + Inet et The ipule given by the eternal force (friction force) in econd i I = F t net et f et = F t There' a inu ign becaue the force i toward the left. The ipule i then I = µf t net et The peed can then be calculated with N = µg t N = kg 9.8 = kg kg 018 Verion 10 - Moentu 43

44 Luc Treblay p = p + I tot tot net et ( ).05kg v = kg kg kg kg.05kg v = kg v = v =.657 kg kg A a changing object, i.e. a rocket epelling ga with an epulion peed of v ep, will now be conidered. Note that v ep i the peed relative to the rocket and not the peed relative to the ground. The ga i epelled at a rate R, which correpond to the nuber of kilogra of ga epelled each econd. reviionworld.co.uk/a-level-level-reviion/phyic/force-otion/oentu-econd-law/oentu-econd-law-0 Thrut Force of a Rocket In the figure to the right, a rocket epel a all aount of ga. The total oentu of the yte (rocket and ga) ut be the ae before and after the ejection. Thi ean that rocket ga rocket ga ( ) ( ep ) rocket rocket rocket 0 = dp v d rocket p = p p + p = p + p p + vd = p + dp + d v v rocket ep ep dp = v d A the force on the rocket i dp rocket /dt, the equation becoe nothingnerdy.wikipace.co/forces+change+momentum 018 Verion 10 - Moentu 44

45 Luc Treblay dp = v d dp dt F rocket rocket thrut = v = v ep ep ep d dt d dt Thi d/dt i the rate at which the a of the rocket change. Thi i actually the rate of a ejection of the ga (R). Thrut Force of a Rocket F = v R thrut ep Speed of the Rocket The force i contant if the epelled ga velocity and rate of a ejection are contant. However, it i tricky to find the final velocity of the rocket becaue the a of the rocket decreae contantly. Thi ean that the acceleration of the rocket i not contant. The acceleration of the rocket i F thrut = Ma F a = M where M i the a of the rocket (which i contantly changing). Since the thrut force i thrut thrut F = v R ep The acceleration i vepr a = M A the a M decreae at rate R, the a a a function of tie i M = Rt where i the initial a of the rocket. The acceleration equation then give 018 Verion 10 - Moentu 45

46 Luc Treblay vepr a = Rt dv R = vep dt Rt R dv = vep dt Rt Integrating fro the initiation of thrut (t = 0) to a tie T give the peed at tie T. v v T R dv = v dt Rt ep 0 v T [ v] = vep ln ( ) v Rt 0 ep ( ln ( ) ln ) v v = v RT v = v + v ep ln RT The peed i then Speed of a Rocket Epelling Ga during Tie T v = v + vep ln RT v = v + vep ln In the econd equation i the a of the rocket at tie T, which i RT. Eaple A rocket, initially at ret in pace, begin to eject ga at v ep = 000 /. The ga i ejected at the rate of 1000 kg/. The initial a of the rocket i 100 ton, which include 60 ton of ga to be ejected. a) What i the thrut force? The force i F thrut = Rv ep kg = =,000,000N 018 Verion 10 - Moentu 46

47 Luc Treblay b) What i the peed of the rocket once all the ga i ejected? The final peed of the rocket i ln v = v + vep 100 ton = ln 40 ton = 1833 c) How long did it take for the rocket to reach thi peed? A 60,000 kg of ga are ejected at a rate of 1000 kg/, the ejection lated 60,000kg t = = kg Eaple At liftoff, a rocket initially at ret tart to eject ga with a peed of v ep = 000 /. The ga i ejected at a rate of 1000 kg/. The initial a of the rocket i 100 ton, including 60 ton of ga to be ejected. a) What i the initial acceleration of the rocket? There are two force eerted on the rocket: the force of gravity and the thrut force of the engine. Therefore, the force equation give g F y = a F + F = a thrut g + Rv = a N kg 100,000kg = 100,000kg a ep kg a = 10. nothingnerdy.wikipace.co/forces+change+momentum A thi acceleration i poitive, it i directed upward, and the rocket anaged to lift off. The acceleration could not be negative (downward). If the rocket failed to lift off, a noral force i added o that a = Verion 10 - Moentu 47

48 Luc Treblay Note that thi 10. /² acceleration i not contant. It increae a the a of the pacecraft decreae. b) What i the peed of the rocket 30 econd after liftoff? Here, the forula of the peed of a rocket cannot be ued directly becaue the gravitational force alo act on the rocket. However, it ay be ued if a ethod unknown up to now i eployed. Up to now, the u of the force wa done in order to find the acceleration and then the velocity of an object. Actually, it i poible to do thi u later in the olution. The acceleration ade be each force can be found, and thee acceleration ay then be added to find the net acceleration and the velocity. It i alo poible to find the acceleration and then the velocity reulting fro the action of each force and then u thee velocitie to obtain the net velocity of the object. Thee ethod all give the ae reult. It doe not really atter if you u the force, the acceleration or the velocitie, provided that the u i done at oe point. Thi trick will be ued here becaue we know the velocity given by each of the force acting on the rocket. The engine thrut force give an upward peed of v ln 1 = v + vep and the gravitational force give a downward peed of v = 9.8 ² t (Since all object fall with an acceleration of 9.8 /².) The reulting peed i v = v + vep ln 9.8 ² t After 30 econd, the a of the rocket i 70 ton (ince 1 ton i ejected each econd). The peed i then v = v + vep ln 9,8 t ² 100 ton = ln 9.8 ² ton = Verion 10 - Moentu 48

49 Luc Treblay Coponent of the Ipule Given to an Object I = F t I y = Fy t Iz = Fz t Net Ipule Given to an Object (I net ) I = I = I + I + I + net 1 3 I = I = I + I + I + y net y y1 y y3 I = I = I + I + I + z net z z1 z z3 Ipule Given by a Variable Force Acting on an Object I = F t I = F t I = F t y y z z Contant F Contant F Contant F Coponent of the Ipule Given by a Variable Force on an Object (Mot General Forula) t t t I = F dt I = F dt I = F dt y y z z t t t The ipule given to an object i the area under the curve of the force acting on the object a a function of tie Moentu ( p ) p = v in coponent, thi i p = v p = v p = v y y z z 018 Verion 10 - Moentu 49

50 Luc Treblay Ipule-Moentu Theore I = p net in coponent, thi i I = p I = p I = p net y net y z net z Relationhip between Force and Moentu (Contant Force) F net p = t Newton Second Law (More General) F net dp = dt Average Force Acting on an Object I p I p I p F = = F = = F = = t t t t t t y y z z y z Moentu Conervation Law p = p if F = 0 tot tot et yte p = p if F = 0 y tot y tot y et yte p = p if F = 0 z tot z tot z et yte Copletely Inelatic Colliion In a copletely inelatic colliion, the two object tick together and have the ae velocity after the colliion. The, only the oentu of the yte i conerved. p = p p y = p y 018 Verion 10 - Moentu 50

51 Luc Treblay Elatic Colliion In an elatic colliion, the oentu and kinetic energy are the ae before and after the colliion. p p E y k = p = p y = E k Relationhip between Moentu and Kinetic Energy E k p = Ipule-Moentu Theore for a Syte p = p + I tot tot net et p = p + I tot y tot y net et y p = p + I tot z tot z net et z where I net et are net ipule ade by eternal force. Thrut Force of a Rocket F = v R thrut ep Speed of a Rocket Epelling Ga during Tie T v = v + vep ln RT v = v + vep ln 018 Verion 10 - Moentu 51

52 Luc Treblay 10.1 Ipule 1. Karee puhe a 10 kg bo with an 800 N horizontal force toward the right for 10 econd. The coefficient of friction between the ground and the bo i of a) What are the coponent of the ipule given by the gravitational force? b) What are the coponent of the ipule given by the noral force? c) What are the coponent of the ipule given by the friction force? d) What are the coponent of the ipule given by the force eerted by Karee?. Gilbert puhe a 30 kg crate uphill on a 0 lope for 0 econd. The crate ha an acceleration of 1 /² toward the top of the lope, and there i a frictional force of F f = 70 N oppoed to the otion of the crate. cn.org/content/4150/latet/?collection=col11406/latet a) What are the coponent of the ipule given by the gravitational force? b) What are the coponent of the ipule given by the friction force? c) What are the coponent of the ipule given by the force eerted by Gilbert? d) What are the coponent of the ipule given by the noral force? e) What are the coponent of the net ipule? 018 Verion 10 - Moentu 5

53 Luc Treblay 3. The following contant force F1 = i + j 4k N ( ) i eerted on an object for 3 econd. Then the following contant force F = 4i + 5 j + k N ( ) i eerted on the object for 5 econd. What are the 3 coponent of the ipule given to the object during thee 8 econd? 4. Here i the graph of the -coponent of a force acting on an object a a function of tie. What i the -coponent of the ipule given to the object between t = 0 and t = 8? 5. The force F = t 9 N ² act on an object. What i the -coponent of the ipule given to the object between t = 0 and t = 5? 018 Verion 10 - Moentu 53

54 Luc Treblay 10. Ipule-Moentu Theore 6. Carole had parked her 000 kg Wetphalia on a lope but the brake failed. Thu, later, he ee her Wetphalia rolling by itelf at 5 / on a horizontal portion of the road. She then run to top her vehicle. She goe in front of the vehicle and trie to top it by eerting a 50 N force. Uing the forula Δ deterine the peed of the Wetphalia after Carole ha eerted the force for 0 econd. c.codanacadey.org/branche/phyicofdriving/inde.php?odule=pageater&page_uer_op=view_page&page_i d=3&mmn_poition=3: A New Verion of Newton Second Law 7. At an archery eeting, Giele launche a 100 g arrow at 150 /. What wa the agnitude of the average force on the arrow if the force i eerted for 0.05 econd on the arrow? 8. A 0 g rifle bullet oving at 900 / i topped by a wooden block. If the bullet top in econd, what i the agnitude of the average force on the bullet? 9. Jutin car drove into a wall and bounce back a little a hown in the figure. What wa the average force acting on the wall if the a of the car i 1150 kg and the colliion lated 0.10? Verion 10 - Moentu 54

55 Luc Treblay 10. A 50 g ball bounce off the ground. The following figure how the velocitie iediately before and iediately after the colliion with the ground. What i the average force (agnitude and direction) eerted on the ball during the colliion if it lat 0.06? 11. Here a force-veru-tie graph of the force acting on an object. What i the average force eerted on the object between t = 0 ec and t = 30? Moentu Conervation 1. Edward hold a ball while he i at ret on ice (iage A). There i no friction between the ice and Edward boot. Edward then launche the ball with a peed of 0 / (iage B). What i Edward velocity after he threw the ball if Edward a i 65 kg and the a of the ball i 800 g? Verion 10 - Moentu 55

56 Luc Treblay 13. A ball i heading at 0 k/h toward Marie-Sophie, who i at ret (iage A). Marie-Sophie i on an icy urface that offer no friction. What i the velocity of Marie-Sophie once he caught the ball (picture B)? Two atronaut in pace far fro any planet are initially at ret. Yuri (80 kg) throw a 0 kg caniter toward Valentina (70 kg) at a peed of 5 /. What are the velocitie of the two atronaut once Valentina ha grabbed the caniter? public.wu.edu/~jtd/phyic05/chap7/chap7_.ht 15. Helut and Brünnhilde are both at ret on an icy urface offering no friction (iage A). Helut then puhe Brünnhilde, giving her a peed of 10 / (iage B). What i the velocity of Helut? Verion 10 - Moentu 56

57 Luc Treblay 16. McPheron i initially at ret at the end of a floating log alo at ret. McPheron then tart walking toward the other end of the log with a peed of.7 /. How long doe it take for McPheron to get to the other end of the log if the a of McPheron i 94 kg, the a of the log i 345 kg, and the length of the log i 5? The HMS Leander wa a Britih warhip active between 1780 and 1813 (although he wa briefly captured by France for a few onth in 1798 and 1799). She wa not a huge warhip a he wa ranked a a fourth-rate veel (like the hip in the figure) but there were till 44 gun on board. During a ide-by-ide fight with another hip, only the gun on one ide of the veel were ued. Thee gun were hooting cannonball, whoe total a wa 180 kg, at the peed of 45 /. What wa the Leander recoil peed when all gun were fired iultaneouly if the a of the hip wa 1,00 ton (which doe not include the cannonball that were fired)? Verion 10 - Moentu 57

58 Luc Treblay 18. A 10 kg cannonball eplode into three fragent. The figure how the peed and direction of the velocity of the three fragent after the eploion. What wa the peed and direction of the velocity of the cannonball before the eploion? In a nuclear alpha decay, the atoic nucleu eject an alpha particle (which i copoed of two neutron and two proton) while releaing energy. The releaed energy i tranferred into the kinetic energie of the two particle (the nucleu and the alpha particle) after the diintegration. What are the velocitie of the two particle after the diintegration knowing that the energy releaed i J and that, after the diintegration, the ae are kg (alpha particle) and kg (nucleu)? cn.org/content/4633/latet/?collection=col11406/latet 018 Verion 10 - Moentu 58

59 Luc Treblay 0. Leon fire the rifle hown in the figure. Initially, the carriage i at ret. The carriage roll without friction on the track. The bullet ha a a of 30 g and a peed of 900 /. What i the recoil velocity of the carriage if the a of the carriage (including everything on it) i 150 kg? One-Dienional Colliion 1. The two train car of the figure are involved in a copletely inelatic colliion. What i the velocity of the car after the colliion? antonine-education.co.uk/page/phyic_gcse/unit_/add_07_moentu/add_page_07.ht. Fabrice and Raphael are two atronaut heading toward each other with the velocitie hown in the figure. When they eet, they cling to each other. What i their velocity after they have clung to each other? Verion 10 - Moentu 59

60 Luc Treblay 3. A 5 kg block (block 1) oving at 10 / collide with another block (block ) oving at /. The peed and the direction of the velocity of the two block are hown in the figure. After a copletely inelatic colliion, the block have a peed of 1 / toward the left. What i the a of block? collide-block--a-310-initial-q In the ituation hown in the figure, find the peed v and v 3 if the block are involved in two copletely inelatic colliion. and-anwer/--i5-proble-block-a-00- oving-140-undergoe-copletely-inelaticcolliion-tation-q In the following ituation, what i the aiu angle reached by the pendulu after the colliion? Verion 10 - Moentu 60

61 Luc Treblay 6. With the conervation of the -coponent of oentu, deterine the velocity of the carriage after the colliion with the ball if the ball ha a a of 10 kg and the carriage ha a a of 00 kg? phyic.tackechange.co/quetion/57164/inelatic-colliion-and-ipule 7. What are the velocitie of thee ball after an elatic colliion? titan.bloofield.edu/factaff/dnicolai/phyic/phyic105/phy105-leon/leon6-105.ht 8. The two ateroid hown in the figure are involved in a colliion (it i not known whether the colliion i elatic or inelatic). After the colliion, the 50 kg ateroid ha a peed of 1 /. a) What i the peed of the 500 kg ateroid after the colliion? b) What fraction of the kinetic energy i lot in the colliion? c) What i the change in oentu of the 500 kg ateroid? d) What i the change in oentu of the 50 kg ateroid? 018 Verion 10 - Moentu 61

62 Luc Treblay 9. A 4 g bullet ove with a peed of 750 /. It then hit an 1150 g wooden block. However, the bullet coe out on the other ide the block with a peed of 30 /. What i the peed of the wooden block after the bullet ha paed through it? In the ituation hown in the figure, the kg block lide on a frictionle urface and then ake a copletely inelatic colliion with a 3 kg block. The two block then lide on a horizontal urface. However, there i oe friction between the block and the horizontal urface. What will be the topping ditance of the block if the friction coefficient i 0.5? phyic.tackechange.co/quetion/4469/elatic-colliion-and-oentu 31. In the following colliion how that the peed after the colliion are given by 1 1 v = v v = v Verion 10 - Moentu 6

63 Luc Treblay 3. In the following colliion how that the kinetic energy lot in the colliion i given by 1 1 Ek = v Two-Dienional Colliion 33. What i the velocity (agnitude and direction) of thee ball after they ake a copletely inelatic colliion? dev.phyiclab.org/docuent.ap?doctype=3&filenae=moentu_moentu.l 34. Here a copletely inelatic colliion between vehicle A, whoe a i 1500 kg, and vehicle B, whoe a i 000 kg. After the colliion, both vehicle tick together and have a peed of 1 / in the direction indicated in the figure (v ). What wa the peed of each car before the colliion? reader.chegg.co/hoework-help/quetion-and-anwer/day-driving-_a-1500-kg-car-going-northeatyouebarraingly-ah-intructor-rquo--_b-q Verion 10 - Moentu 63

64 Luc Treblay 35. In the following elatic colliion, deterine the peed of the 3 kg ball and the velocity (agnitude and direction) of the 1 kg ball Here a colliion (it i not known whether the colliion i elatic or inelatic). canu.ucalgary.ca/ap/clae/info/ualberta/colliion_d/applethelp/leon/leon_1.htl a) What i the velocity of ball (agnitude and direction) after the colliion? b) How uch kinetic energy i lot during thi colliion? c) I the colliion elatic or inelatic? 018 Verion 10 - Moentu 64

65 Luc Treblay 10.8 Ipule-Moentu Theore for a Syte 37. In the ituation hown in the figure, a 50 N force continuouly puhe the 10 kg block which i initially at ret. One econd later, the block ake a copletely inelatic colliion with a 0 kg block. What i the velocity of the block econd after the colliion? (There i no friction between the block and the ground.) collide-block--a-310-initial-q A 0 g bullet hit a 1 kg block with a peed of 500 / a hown in the figure. After the colliion, the ball i lodged in the block. What i the velocity of the block 0.5 econd after the colliion? a-block-a--i-q Rocket Propulion 39. The rocket Ariane 5 eject ga at a peed of 300 / and at the rate of 3400 kg/. The initial a of the rocket i 710,000 kg, including fuel. a) What i the thrut force of the engine? b) Aue that thi rocket i initially at ret in pace. What will be it peed if the engine i fired for a inute? c) Now aue the rocket i fired upward fro the urface of the Earth. What will be the peed of the rocket 30 econd after the engine wa fired? 018 Verion 10 - Moentu 65

66 Luc Treblay 40. When the engine of a 100-ton rocket are fired for 60 econd, the peed of the rocket in pace change fro 15 k/ to 18 k/. What i the ga ejection peed if 0.75 ton of fuel i ejected per econd? Challenge (Quetion ore difficult than the ea quetion.) 41. In the copletely elatic colliion hown in the figure, what will be the aiu pring copreion if there i no friction? 4. Show that in a copletely elatic colliion between two ball the ae a uch a the one hown in the figure, the angle between the trajectorie of the ball after the colliion i alway 90. (Before the colliion, the white ball ha oe peed and the black ball i at ret.) Verion 10 - Moentu 66