5.5. Collisions in Two Dimensions: Glancing Collisions. Components of momentum. Mini Investigation

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1 Colliion in Two Dienion: Glancing Colliion So ar, you have read aout colliion in one dienion. In thi ection, you will exaine colliion in two dienion. In Figure, the player i lining up the hot o that the cue all (the white all) will hit another illiard all at an angle, directing it toward the corner pocket. What coponent o the cue all oentu will e tranerred to the target all i the hot i ucceul? The law o conervation o oentu and conervation o kinetic energy will apply jut a they did or one-dienional interaction. However, to calculate oentu or two-dienional prole, conider the x-coponent and y-coponent o orce and otion independently. 5.5 Figure Bil liard require player to ater the ue o glancing colliion. Mini Invetigation glancing Colliion Mini Invetigation Skill: Peroring, Oerving, Analyzing, Counicating In thi invetigation, you will odel glancing colliion to oerve and analyze how they work. Equipent and Material: eye protection; air tale; puck; arle; illiard all. Put on your eye protection. Set up an air tale with two puck. When you unplug the air tale, pull the plug and not the cord. Wear cloed-toe hoe to protect your eet in cae the puck l ie o the tale. Puh the oject lightly and cautiouly.. Puh the i rt puck toward the econd to caue a gentle head-on colliion. Oerve the change in the peed o the puck ater the colliion. SKILLS HANDBOOK a. 3. Repeat thi proce, ut vary the angle o colliion and the initial peed o the i rt puck. Oerve the change in the peed o the puck and the direction o their i nal velocitie. 4. Try the ae invetigation uing dierent oject, uch a arle or illiard all. A. How doe the peed o the econd puck copare to the initial peed o the i rt puck a you vary the angle o colliion? K/u T/I a B. How did uing arle or illiard all aect the change in direction and velocity? How wa thi dierent ro uing the puck? Briel y uarize your oervation. T/I a Coponent o oentu Dealing with colliion in two dienion involve the ae aic idea a dealing with colliion in one dienion. Now, however, the inal velocity o each oject involve two unknown: the two coponent o the velocity vector. For oject in otion in two dienion, the change in oentu or each coponent can e conidered independently: SF > x Dt 5 Dp > x SF > ydt 5 Dp > y Siilarly, the conervation o oentu equation can e expreed in ter o horizontal and vertical coponent: p > i x p > i x 5 p > x p > x p > i y p > i y 5 p > y p > y 5.5 Colliion in Two Dienion: Glancing Colliion 49

2 glancing colliion a colliion in which the irt oject, ater an ipact with the econd oject, travel at an angle to the direction it wa originally travelling Conider the colliion o two illiard all hown in Figure. In thi hot, the cue all () will collide with the target all (), initially at ret, ending it at an angle toward the corner pocket, and the cue all will continue travelling at an angle u ro it original direction o travel. Both oject are travelling at an angle to the direction o their original coure. Thi type o colliion i called a glancing colliion. In the ollowing Tutorial, we ue coponent to analyze the phyic o a glancing colliion. v y x at ret v x v y Invetigation 5.5. v x? Conervation o Moentu in Two Direction (page 60) Ater learning aout the phyic o glancing colliion, peror Invetigation 5.5. to explore how oentu i conerved in colliion that occur in two dienion. y () Figure A cue all triking another all at an angle caue oth all to change direction. u x Tutorial Analyi o Glancing Colliion In thee Saple Prole, you will apply conervation o oentu in two dienion. Saple Prole : Analyi o a Glancing Colliion In a gae o curling, a colliion occur etween two tone o equal a. The oject tone i initially at ret. Ater the colliion, the tone that i thrown ha a peed o 0.56 / in oe direction, repreented y u in Figure 3. eore the colliion The oject tone acquire a velocity v > / at an angle o ro the original direction o otion o the thrown tone. Deterine the initial velocity o the thrown tone. ater the colliion p y p in u y v p x p co u i u x x p x p co () p y p in Figure 3 The curling tone collide with the oject tone in a glancing colliion. () Both curling tone ove o in dierent direction. We can analyze their inal velocitie to deterine the initial velocity o the thrown curling tone. v 50 Chapter 5 Moentu and Colliion

3 Given: 5 ; v > i 5 0 /; v > /; v > /; Required: v > i Analyi: Chooe a coordinate yte to identiy direction: let poitive x e to the right and negative x e to the let. Let poitive y e up and negative y e down. Apply conervation o oentu independently in the x-direction and the y-direction. Begin y applying conervation o oentu in the y-direction to deterine the direction o the inal velocity o the thrown tone. Then apply conervation o oentu in the x-direction to calculate the initial velocity o the thrown tone. Solution: In the y-direction, the total oentu eore and ater the colliion i zero. p Tiy 5 p Ty 5 0 Thereore, ater the colliion: y v y 5 0 Divide oth ide y and utitute the vertical coponent o each velocity vector. Note that the vertical coponent o the irt tone velocity i directed up, o it value i poitive, wherea the vertical coponent o the econd tone velocity i directed down, o it value i negative. y v y 5 0 in u v in 5 0 Rearrange thi equation to iolate in u. in u 5 v in in u 5 v in Sutitute the given value and olve or in u. in u / in / in u Apply the invere ine to oth ide to olve or u. u 5 in u 5.08 one extra digit carried The irt tone i travelling at an angle o aove the horizontal ater the colliion. Now ue conervation o oentu in the x-direction to olve or the initial peed o the thrown tone. p Tix 5 p Tx Note that the oject tone i at ret eore the colliion, o it initial oentu i zero. v ix v ix 5 x v x Divide oth ide o the equation y. v ix 5 v v x x v ix 5 x v x Sutitute the known value or the inal horizontal velocity coponent o the tone. Note that all vector are directed to the right, o all velocitie are poitive. v ix 5 co u v co co.08 / 0.4 co 308 / / / one extra digit carried v ix / Stateent: The initial velocity o the thrown tone i 0.88 / [right]. Saple Prole : Inelatic Glancing Colliion Two cro-country kier are kiing to a croing o horizontal trail in the wood a hown in Figure 4. Skier i travelling eat and ha a a o 84 kg. Skier i travelling north and ha a a o 7 kg. Both kier are travelling with an initial peed o 5. /. One o the kier orget to look, reulting in a right-angle colliion with the ki locked together ater the colliion. Calculate the inal velocity o the two kier. v i North v y v x v Eat v i Figure Colliion in Two Dienion: Glancing Colliion 5

4 Given: inelatic colliion; 5 84 kg; 5 7 kg; v > i 5 5. / [E]; v > i 5 5. / [N] Required: v > Analyi: According to the law o conervation o oentu, p > T i 5 p > T. Since the initial velocitie are at right angle to each other, a hown in Figure 5, you can calculate the total velocity and oentu uing the Pythagorean theore and trigonoetry: p 5 p p, and tan u 5 a p p north p O Figure 5 p p p p eat Solution: The irt kier oentu i p > 5 v > 5 84 kg a5. 3E4 p > 5 48 kg# / 3E4 one extra digit carried The econd kier oentu i p > 5 v > 5 7 kg a5. 3N4 p > kg# / 3N4 one extra digit carried The agnitude o the total oentu can e calculated y applying the Pythagorean theore: p 5 p p p 5 "p p 5 a48 kg # kg# a367 Å p kg# / one extra digit carried The direction can e deterined y applying the tangent ratio: tan u 5 a p p u 5 tan a p p 48 kg # 5 tan ± kg# 367 u The direction o the two kier i [N 49 E]. Conervation o oentu tell u that the inal total oentu o the kier ut equal thi initial oentu. Since the colliion i perectly inelatic, oth kier have the ae inal velocity: p > 5 v > v > 5 v > v > p > kg # 5 3N 498 E4 84 kg 7 kg v > / 3N 498 E4 Stateent: Ater the colliion, the kier are travelling together with a velocity o 3.6 / [N 49 E]. Practice. Two reight train have a copletely inelatic colliion at a track croing. Engine ha a a o kg and i initially travelling at 45 k/h [N]. Engine ha a a o kg and i initially travelling at 53 k/h [W]. Calculate the inal velocity. T/I A [an: 9.7 / [N 5 W]. A tar o a kg oving with a velocity o / [E] collide with a econd tar o a kg oving with a velocity o / at a right angle to the path o the irt tar. I the two join together, what i their coon velocity? T/I A [an: / [58 to the initial path o the econd tar]] 5 Chapter 5 Moentu and Colliion

5 5.5 Review Suary The law o conervation o oentu and conervation o kinetic energy or colliion in two dienion are the ae a they are or onedienional colliion. Moentu i conerved or elatic and inelatic colliion. Kinetic energy i conerved only in elatic colliion. The act that oentu i a vector quantity ean that prole involving two-dienional colliion can e olved y independently analyzing the x-coponent and y-coponent. Quetion. Two all o equal a undergo a colliion (ee Figure 6). Ball i initially travelling horizontally with a peed o 0.0 /, and all i initially at ret. Ater the colliion, all ove away with a velocity o 4.7 / at an angle o u ro it original path and all ove away at an unknown angle. Deterine the agnitude and direction o velocity o all ater the colliion. K/u T/I () Figure 6 v i 0.0 / 4.7 / u 60.0 v i 0 / v?. A hockey puck o a 0.6 kg, liding on a nearly rictionle urace o ice with a velocity o.0 / [E], trike a econd puck at ret with a a o 0.7 kg. The irt puck ha a velocity o.5 / [N 3 E] ater the colliion. Deterine the velocity o the econd puck ater the colliion. T/I a 3. Two hockey puck o equal a approach each other. Puck ha an initial velocity o 0.0 / [S 45 E], and puck ha an initial velocity o 5 / [S 45 W]. Ater the colliion, the irt puck i oving with a velocity o 0.0 / [S 45 W]. K/u T/I C Deterine the inal velocity o the econd puck. () I thi colliion elatic, perectly inelatic, or (non-perectly) inelatic? Explain your reaoning. 4. An autooile collide with a truck at an interection. The car, o a kg, i travelling at 3 k/h [S]; the truck ha a a o kg and i travelling at 48 k/h [E]. The colliion i perectly inelatic. Deterine their velocity jut ater the colliion. T/I 5. Two all o equal a undergo a colliion. One all i initially tationary. Ater the colliion, the velocitie o the all ake angle o 5.5 and 45.9 relative to the original direction o otion o the oving all. T/I C Draw and lael a diagra to how the all eore and ater the colliion. Lael the angle u and. () Calculate the inal peed o the all i the initial all had a peed o 3.63 /. 6. A caron-4 nucleu, initially at ret, undergoe a nuclear reaction known a eta decay. The nucleu eit two particle horizontally: one with oentu kg. / [E] and another with oentu kg. / [S]. T/I a Calculate the direction o the otion o the nucleu iediately ollowing the reaction. () Deterine the inal oentu o the nucleu. (c) The a o the reidual caron-4 nucleu i kg. Deterine it inal velocity. 7. A neutron o a kg, travelling at. k/, hit a tationary heliu nucleu o a kg. Ater the colliion, the velocity o the heliu nucleu i 0.53 k/ at 5 to the original direction o otion o the neutron. Deterine the inal velocity o the neutron. T/I 8. Your claate ake the ollowing tateent: For a head-on elatic colliion etween two oject o equal a, the ater-colliion velocitie o the oject are at right angle to each other. Evaluate the accuracy o thi tateent. K/u T/I a 5.5 Colliion in Two Dienion: Glancing Colliion 53