4 Conservation of Momentum

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1 hapter 4 oneration of oentu 4 oneration of oentu A coon itake inoling coneration of oentu crop up in the cae of totally inelatic colliion of two object, the kind of colliion in which the two colliding object tick together and oe off a one. The itake i to ue coneration of echanical energy rather than coneration of oentu. One way to recognize that oe echanical energy i conerted to other for i to iagine a pring to be in between the two colliding object uch that the object copre the pring. Then iagine that, jut when the pring i at axiu copreion, the two object becoe latched together. The two object oe off together a one a in the cae of a typical totally inelatic colliion. After the colliion, there i energy tored in the copreed pring o it i clear that the total kinetic energy of the latched pair i le than the total kinetic energy of the pair prior to the colliion. There i no pring in a typical inelatic colliion. The echanical energy that would be tored in the pring, if there wa one, reult in peranent deforation and a teperature increae of the object inoled in the colliion. The oentu of an object i a eaure of how hard it i to top that object. The oentu of an object depend on both it a and it elocity. onider two object of the ae a, e.g. two baeball. One of the i coing at you at 0 ph, and the other at 00 ph. Which one ha the greater oentu? Anwer: The fater baeball i, of coure, harder to top, o it ha the greater oentu. Now conider two object of different a with the ae elocity, e.g. a Ping-Pong ball and a cannon ball, both coing at you at 5 ph. Which one ha the greater oentu? The cannon ball i, of coure, harder to top, o it ha the greater oentu. The oentu p of an object i equal to the product of the object a and elocity : p 4- oentu ha direction. It direction i the ae a that of the elocity. In thi chapter we will liit ourele to otion along a line otion in one dienion. Then there are only two direction, forward and backward. An object oing forward ha a poitie elocity/oentu and one oing backward ha a negatie elocity/oentu. In oling phyic proble, the deciion a to which way i forward i typically left to the proble oler. Once the proble oler decide which direction i the poitie direction, he ut tate what her choice i thi tateent, often ade by ean of notation in a ketch, i an iportant part of the olution, and tick with it throughout the proble. The concept of oentu i iportant in phyic becaue the total oentu of any yte reain contant unle there i a net tranfer of oentu to that yte, and if there i an ongoing oentu tranfer, the rate of change of the oentu of the yte i equal to the rate at which oentu i being tranferred into the yte. A in the cae of energy, thi ean that one can ake prediction regarding the outcoe of phyical procee by ean of Thi claical phyic expreion i alid for peed all copared to the peed of light c /. The relatiitic expreion for oentu i p / c. At peed that are ery all copared to the peed of light, the claical phyic expreion p i a fantatic approxiation to the relatiitic expreion. 0

2 hapter 4 oneration of oentu iple accounting bookkeeping procedure. The cae of oentu i coplicated by the fact that oentu ha direction, but in thi initial encounter with the coneration of oentu you will deal with cae inoling otion along a traight line. When all the otion i along one and the ae line, there are only two poible direction for the oentu and we can ue algebraic ign plu and inu to ditinguih between the two. The principle of oneration of oentu applie in general. At thi tage in the coure howeer, we will conider only the pecial cae in which there i no net tranfer of oentu to or fro the yte fro outide the yte. oneration of oentu in One Dienion for the Special ae in which there i No Tranfer of oentu to or fro the Syte fro Outide the Syte In any proce inoling a yte of object which all oe along one and the ae line, a long a none of the object are puhed or pulled along the line by anything outide the yte of object it okay if they puh and pull on each other, the total oentu before, during, and after the proce reain the ae. The total oentu of a yte of object i jut the algebraic u of the oenta of the indiidual object. That adjectie "algebraic" ean you hae to pay careful attention to the plu and inu ign. If you define "to the right" a your poitie direction and your yte of object conit of two object, one oing to the right with a oentu of kg / and the other oing to the left with oentu 5 kg /, then the total oentu i kg / 5 kg / which i 7 kg /. The plu ign in the final anwer ean that the total oentu i directed to the right. Upon reading thi election you'll be expected to be able to apply coneration of oentu to two different kind of procee. In each of thee two clae of procee, the yte of object will conit of only two object. In one cla, called colliion, the two object bup into each other. In the other cla, anti-colliion the two object tart out together, and pring apart. Soe further breakdown of the colliion cla i pertinent before we get into exaple. The two extree type of colliion are the copletely inelatic colliion, and the copletely elatic colliion. Upon a copletely inelatic colliion, the two object tick together and oe off a one. Thi i the eay cae ince there i only one final elocity becaue they are tuck together, the two object obiouly oe off at one and the ae elocity. Soe echanical energy i conerted to other for in the cae of a copletely inelatic colliion. It would be a big itake to apply the principle of coneration of echanical energy to a copletely inelatic colliion. echanical energy i not conered. The word "copletely inelatic" tell you that both object hae the ae elocity a each other after the colliion. In a copletely elatic colliion often referred to iply a an elatic colliion, the object bounce off each other in uch a anner that no echanical energy i conerted into other for in the colliion. Since the two object oe off independently after the colliion there are two final elocitie. If the ae and the initial elocitie are gien, coneration of oentu yield one equation with two unknown naely, the two final elocitie. Such an equation

3 hapter 4 oneration of oentu cannot be oled by itelf. In uch a cae, one ut apply the principle of coneration of echanical energy. It doe apply here. The expreion "copletely elatic" tell you that coneration of echanical energy doe apply. In applying coneration of oentu one firt ketche a before and an after picture in which one define ybol by labeling object and arrow indicating elocity, and define which direction i choen a the poitie direction. The firt line in the olution i alway a tateent that the total oentu in the before picture i the ae a the total oentu in the after picture. Thi i typically written by ean an equation of the for: p p 4- The Σ in thi expreion i the upper cae Greek letter iga and i to be read the u of. Hence the equation read: The u of the oenta to the right in the before picture i equal to the u of the oenta to the right in the after picture. In doing the u, a leftward oentu count a a negatie rightward oentu. The arrow ubcript i being ued to define the poitie direction. Exaple Now let' get down to oe exaple. We'll ue the exaple to clarify what i eant by colliion and anti-colliion; to introduce one ore concept, naely, relatie elocity oetie referred to a uzzle elocity; and of coure, to how the reader how to apply coneration of oentu.

4 hapter 4 oneration of oentu Exaple 4- Two object oe on a horizontal frictionle urface along the ae line in the ae direction which we hall refer to a the forward direction. The trailing object of a.0 kg ha a elocity of 5 / forward. The leading object of a 3. kg ha a elocity of / forward. The trailing object catche up with the leading object and the two object experience a copletely inelatic colliion. What i the final elocity of each of the two object? EFORE AFTER 5.0 kg 3. kg Σp p p Σp p. 0 kg 5 / 3. kg /. 0kg 3. kg The final elocity of each of the object i 3 forward. 3

5 hapter 4 oneration of oentu 4 Exaple 4-: A cannon of a, reting on a frictionle urface, fire a ball of a. The ball i fired horizontally. The uzzle elocity i. Find the elocity of the ball and the recoil elocity of the cannon. NOTE: Thi i an exaple of an anti-colliion proble. It alo inole the concept of relatie elocity. The uzzle elocity i the relatie elocity between the ball and the cannon. It i the elocity at which the two eparate. If the elocity of the ball relatie to the ground i to the right, and the elocity of the cannon relatie to the ground i to the left, then the elocity of the ball relatie to the cannon, alo known a the uzzle elocity of the ball, i. In cae not inoling gun or cannon one typically ue the notation rel for "relatie elocity" or, relating to the exaple at hand, for "elocity of the ball relatie to the cannon." 0 Σp Σp 0 0 Now ubtitute thi reult into equation aboe. Thi yield: EFORE AFTER Alo, fro the definition of uzzle elocity: Subtituting thi reult into equation yield:

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