AP Phyic Moentu AP Wrapup There are two, and only two, equation that you get to play with: p Thi i the equation or oentu. J Ft p Thi i the equation or ipule. The equation heet ue, or oe reaon, the ybol J or ipule (Mr. Swanon ha neer een thi anywhere ele. Oh well.) Here i what you are uppoed to be able to do. A. Syte o Particle, Linear Moentu 1. Ipule and Moentu: You hould undertand ipule and linear oentu o you can: a. Relate a, elocity, and linear oentu or a oing body, and calculate the total linear oentu o a yte o bodie. Jut ue the good old oentu equation. b. Relate ipule to the change in linear oentu and the aerage orce acting on a body. Jut ue the ipule equation. We did eeral o thee proble. I you hae a colliion, tie, and a orce in a proble, think ipule.. Coneration o Linear Moentu, Colliion a. You hould undertand linear oentu coneration o you can: (1) Identiy ituation in which linear oentu, or a coponent o the linear oentu, i conered. The oentu o an iolated yte (no outide orce) i conered in any interaction. Period. () Apply linear oentu coneration to deterine the inal elocity when two bodie that are oing along the ae line, or at right angle, collide and tick together, and calculate how uch kinetic energy i lot in uch a ituation. Thi i the good old inelatic colliion proble. We e done a bunch. I the colliion happen at right angle, then you are going to hae to look at coponent o 147
oentu (which you can do ince oentu i a ector, right?) Anyway, the idea i that ater the colliion both bodie hae the ae elocity. Thereore: 11 1 I the two bodie are oing at right angle, then it a bit ore coplicated. The ertical oentu i conered and the horizontal oentu i conered, o you can write equation or the coneration o oentu in the x and y direction. Then ue thee to ole or whateer unknown you e been preented with. (3) Analyze colliion o particle in one or two dienion to deterine unknown ae or elocitie, and calculate how uch kinetic energy i lot in a colliion. Pretty uch all the proble that you will ee will inole inding one unknown elocity. Body A oing at elocity ha an eleatic colliion with body A which i at ret. Body B end up with a elocity o, &tc. That ort o thing. Let look at a ew tet proble. The irt one we ll look at i o 001 tet. An incident ball A o a 0.10 kg i liding at 1.4 / on the horizontal tabletop o negligible riction hown aboe. It ake a head-on colliion with a target ball B o a 0.50 kg at ret at the edge o the table. A a reult o the colliion, the incident ball rebound, liding backward at 0.70 / iediately ater the colliion. (a) Calculate the peed o the 0.50 kg target ball iediately ater the colliion. We ue coneration o oentu to ole thi one. ' ' ' ' ' 11 11 11 11 1.0 kg 1.4 0.70 ' 0.4 0.50 kg 1 1 1 ' The tabletop i 1.0 aboe a leel, horizontal loor. The target ball i projected horizontally and initially trike the loor at a horizontal diplaceent d ro the point o colliion. 148
(b) Calculate the horizontal diplaceent d. Thi i a projectile otion proble. We know the ball horizontal elocity and the height o the table, o we can eaily ind the horizontal ditance it trael a it all. 1 y 1. Tie to all: y at t 0.495 a 9.8 The ball ha a horizontal elocity o 0.4 / (which we jut igured out), o the ditance d i iply: xt 0.4 0.495 0.08 In another experient on the ae table, the target ball B i replaced by target ball C o a 0.10 kg. The incident ball A again lide at 1.4 /, a hown aboe let, but thi tie ake a glancing colliion with the target ball C that i at ret at the edge o the table. The target ball C trike the loor at point P, which i at a horizontal diplaceent o 0.15 ro the point o the colliion, and at a horizontal angle o 30 ro the + x-axi, a hown aboe right. (c) Calculate the peed o the target ball C iediately ater the colliion. Thi ound ery hard, angle and all that tu, right? Except we know the ditance it traeled (1.5 ) and we know how long it i in the air beore it hit ae a the preiou proble tie. So thi i a ridiculouly iple proble. Uing the horizontal ditance and the tie to all we can ind the horizontal elocity, which i the elocity it began with, which i the elocity right ater the collion. Find : 0.15 x 0.30 t 0.495 (d) Calculate the y-coponent o incident ball A' oentu iediately ater the colliion. 149
We know that oentu i conered in the x and y direction. So we can u oentu in the y direction. We know that thi oentu ha to add up to be zero a the ball A had no initial oentu in the y direction. 0 in a ay c cy a ay c c ay c c in a ay o 0.10 kg 0.30 in30 0.10 kg 0.15 Fro 1996: Two identical object A and B o a M oe on a one-dienional, horizontal air track. Object B initially oe to the right with peed o. Object A initially oe to the right with peed 3 o, o that it collide with object B. Friction i negligible. Expre your anwer to the ollowing in ter o M and o. a. Deterine the total oentu o the yte o the two object. We iply add up the oentu o each object to get the total oentu. p M 3 M 4M tot A A B B 0 0 0 b. A tudent predict that the colliion will be totally inelatic (the object tick together on colliion). Auing thi i true, deterine the ollowing or the two object iediately ater the colliion. i. The peed. Thi i an inelatic colliion. It i a iple thing to ole: A A B B A B A A A B B B M 3 0 M0 4M0 M M M 0 150
ii. The direction o otion (let or right). The cobined object oe right. All alue are poitie. When the experient i perored, the tudent i urpried to obere that the object eparate ater the colliion and that object B ubequently oe to the right with a peed.5 0. c. Deterine the ollowing or object A iediately ater the colliion. i. The peed o object A ater the colliion. ' ' ' ' A A B B A A B B A A B B B B A A A A B B B A ' B' A 3.5 M M M 0 0 0 M A ' 3.5 1.5 0 0 0 0 ii. The direction o otion (let or right). Object A oe right. All alue are poitie. d. Deterine the kinetic energy diipated in the actual experient. The kinetic energy diipated i the change in kinetic energy. So we ole thi quetion by inding the kinetic energy beore the colliion and the kinetic energy ater the colliion. The dierence in the two i the energy diipated. 1 1 1 1 K K ' ' i A B A B 1 1 1 1 K M M K M M 3.5 1.5 i o o o o i 5 o 4.5 o K M K M diipated i 5 o 4.5 o 0.75 o K K K M M M Fro 199: 151
A 30-kilogra child oing at 4.0 eter per econd jup onto a 50-kilogra led that i initially at ret on a long, rictionle, horizontal heet o ice. a. Deterine the peed o the child-led yte ater the child jup onto the led. You iediately recognize that thi i an inelatic colliion. 30 kg 4 11 111 1.5 30 kg 50 kg 1 b. Deterine the kinetic energy o the child-led yte ater the child jup onto the led. 1 1 1 K 1 80 kg 1.5 90 J Ater coating at contant peed or a hort tie, the child jup o the led in uch a way that he i at ret with repect to the ice. c. Deterine the peed o the led ater the child jup o it. 1 1 30 kg 50 kg 1.5.4 50 kg d. Deterine the kinetic energy o the child-led yte when the child i at ret on the ice. 1 1 Ktotal 50 kg.4 144 J e. Copare the kinetic energie that were deterined in part (b) and (d). I the energy i greater in (d) than it i in (b), where did the increae coe ro? I the energy i le in (d) than it i in (b), where did the energy go? The child ut decelerate in order to top. So a orce negatie to the otion o the led i applied. Thi orce i applied through a ditance reulting in negatie work. The led receie an equal and oppoite (poitie) orce through a ditance reulting in poitie work. Work i added to the led, and work i a change in energy (Work energy theore). So the energy o the led i increaed. Fro 1995: 15
A hown, a 0.0-kilogra a i liding on a horizontal, rictionle air track with a peed o 3.0 eter per econd when it intantaneouly hit and tick to a 1.3-kilogra a initially at ret on the track. The 1.3-kilogra a i connected to one end o a ale pring, which ha a pring contant o 100 Newton per eter. The other end o the pring i ixed. a. Deterine the ollowing or the 0.0-kilogra a iediately beore the ipact. i. It linear oentu. p 0.0 3.0 0.60 kg p kg ii. It kinetic energy. 1 K K 1 0.0 kg 3.0 0.90 J b. Deterine the ollowing or the cobined ae iediately ater the ipact. i. The linear oentu p ii. kg pi p 0.60 The kinetic energy p 1 p 0.60 kg 1 0.0 kg 1.3 kg 0.40 1 1 K 1 0.0 kg 1.3 kg 0.4 0.1 J Ater the colliion, the two ae undergo iple haronic otion about their poition at ipact. c. Deterine the aplitude o the haronic otion. 153
T 1 U K kx K 0.1 N x k N 100 d. Deterine the period o the haronic otion. 0.0 kg 1.3 kg 0.77 k kg 100 0.050 Fro 1994: A track conit o a rictionle arc XY, which i a quarter-circle o radiu R, and a rough horizontal ection YZ. Block A o a M i releaed ro ret at point X, lide down the cured ection o the track, and collide intantaneouly and inelatically with identical block B at point Y. The two block oe together to the right, liding pat point P, which i a ditance l ro point Y. The coeicient o kinetic riction between the block and the horizontal part o the track i. Expre your anwer in ter o M, l,, R, and g. a. Deterine the peed o block A jut beore it hit block B. 1 Ui K gh gh gr b. Deterine the peed o the cobined block iediately ater the colliion. M gr 1 pi p 111 gr M M c. Deterine the aount o kinetic energy lot due to the colliion. 1 1 gr K KBeore KAter M gr M 1 gr MgR K M gr M MgR 1 MgR 4 154
d. The peciic heat o the aterial ued to ake the block i c. Deterine the teperature rie that reult ro the colliion in ter o c and the other gien quantitie. (Aue that no energy i tranerred to the track or to the air urrounding the block.) Q 1 1 gr QcT T MgR c Mc 4c e. Deterine the additional theral energy that i generated a the block oe ro Y to P Work done by riction = heat generated W N W N Mg l Mgl 155