LINEAR MOMENTUM Imagne beng on a skateboad, at est that can move wthout cton on a smooth suace You catch a heavy, slow-movng ball that has been thown to you you begn to move Altenatvely you catch a lght, ast movng ball you also move n the same way Catchng a lghtweght ball movng ast enough has the same eect as catchng a slow movng heavy ball Such obsevatons lead to the denton o a quantty called lnea momentum, p, and s dened as the poduct o the mass m and the velocty v o an object p = mv unts: kg.m/s I the heavy ball has twce the mass o the lght ball, but the lght ball has twce the speed o the heavy ball, the momenta o the two ae equal n magntude The above equaton s only vald o objects movng n a staght lne Because velocty s a vecto (magntude and decton), so too s the momentum 1
CHANGE IN MOMENTUM Consde (a) a 0.01kg toy bea dopped to the loo, httng wth a speed o 4.0m/s and does not bounce; (b) a 0.01kg ubbe ball dops, hts the loo at 4.0m/s, but bounces upwad Assumng an deal ubbe ball, ts ntal upwad speed s 4.0m/s The coodnate system s ntoduced as above: t s seen that nethe object has momentum n x decton Only consde y component, p y Bea: p y, = m(-v); p y, = m(0) = 0 Bea: p y = p y, - p y, = 0 m(-v) = mv = 0.4kg m/s Ball: p y, = m(-v); p y, = mv Ball: p y = p y, - p y, = mv m(-v) = 2mv = 0.8kg m/s Vecto sum o momenta: ptotal = p1 + p2 + p3 + K 2
ADDING MOMENTA: EXAMPLE A peson thows some bead nto a duck pond. Two 4.0kg ducks and a 9.0kg goose paddle apdly towad the bead. I the ducks swm at 1.1m/s, and the goose at 1.3m/s, nd the magntude and decton o the total momentum o the thee bds. 3
MOMENTUM AND NEWTON S SECOND LAW Remembe Newton s 2 nd Law: Expesson only vald o objects wth constant mass Moe geneal law whch holds even the mass changes s F = The net oce actng on an object s equal to the change n ts momentum dvded by the tme nteval dung whch the change occus The net oce s the ate o change o momentum wth tme Snce p = p p = m v m v I mass s constant, then m = m = m Thus p = m v m v = m( v v ) = m v Snce Acceleaton: ate o change o velocty wth tme Theeoe p F = p = m v F = p = ma F = ma v a = v 4
CONSERVATION OF LINEAR MOMENTUM Recall that F = p p = Reaangng gves F ( ) I the net oce actng on an object s zeo: Its change n momentum s also zeo: F = 0 p = F = ( ) 0 Expessng the change o momentum n tems o ts ntal and nal values: p = p p = 0 O p = p Snce the momentum does not change when the sum o the net oces actng on an object s zeo, the expesson used s that momentum s conseved 5
INTERNAL VERSUS EXTERNAL FORCES (1) Hee systems composed o moe than one object ae consdeed The net oce actng on a system o objects s the sum o oces appled om outsde the system (extenal oces) and oces actng between the objects wthn the system (ntenal oces) Fnet = F = Fext + Fnt Consde two canoes loatng at est next to one anothe Consde the system as the two canoes and the people nsde them A peson n canoe 1 pushes on canoe 2, exetng a oce F on canoe 2 2 Newton s 3 d Law says that thee wll be an equal and opposte oce: F 1 = F 2 - both ae ntenal oces and they sum to zeo 6
INTERNAL VERSUS EXTERNAL FORCES (2) Intenal oces, lke all oces, always occu n actoneacton pas Because the oces n acton-eacton pas ae equal and opposte (3 d Law), ntenal oces must always sum to zeo Thus the net oce actng on a system o objects s smply the sum o the extenal oces actng on t Fnet = F = Fext + Fnt = Fext The extenal oces may o may not sum to zeo In the canoe example, the extenal oces ae the weghts o the people and the canoes actng downwad, the upwad (nomal) oce exeted by the wate to keep the canoes aloat These oces sum to zeo, and thee s no acceleaton n the vetcal decton Because the ntenal oces cancel, the change n net momentum s thus p net = ( Fext ) Intenal oces have absolutely no eect on the net momentum o a system I the net extenal oce actng on a system s zeo, ts net momentum s conseved p1, + p2, + p3, + K = p1, + p2, + p3, + K 7
COMPARING VELOCITY AND MOMENTUM: EXAMPLE Two goups o canoests meet n the mddle o a lake. Ate a be vst, a peson n canoe 1 pushes on canoe 2 wth a oce o 46N to sepaate the canoes. I the mass o canoe 1 and ts occupants s 130kg, and the mass o canoe 2 and ts occupants s 250kg, nd the momentum o each canoe ate 1.2s o pushng. Hnt you may nd t useul to use v x = v 0x + a x t 8