Momentum. Momentum. Impulse. Momentum and Collisions
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1 Momentum Momentum and Collsons From Newton s laws: orce must be present to change an object s elocty (speed and/or drecton) Wsh to consder eects o collsons and correspondng change n elocty Gol ball ntally at rest, so some o the KE o club transerred to prode moton o gol ball and ts change n elocty Method to descrbe s to use concept o lnear momentum Lnear momentum = product o mass elocty scalar ector Momentum Impulse Vector quantty, the drecton o the momentum s the same as the elocty s Apples to two-dmensonal moton as well p x m p m x and p y m Sze o momentum: depends upon mass depends upon elocty y In order to change the momentum o an object (say, gol ball), a orce must be appled The tme rate o change o momentum o an object s equal to the net orce actng on t F net p m( ) ma or : p F t t t Ges an alternate statement o Newton s second law (F t) s dened as the mpulse Impulse s a ector quantty, the drecton s the same as the drecton o the orce net
2 Graphcal Interpretaton o Impulse Example: Impulse Appled to Auto Collsons Usually orce s not constant, but tme-dependent mpulse F t area under F() t cure t I the orce s not constant, use the aerage orce appled The aerage orce can be thought o as the constant orce that would ge the same mpulse to the object n the tme nteral as the actual tme-aryng orce ges n the nteral I orce s constant: mpulse = F t The most mportant actor s the collson tme or the tme t takes the person to come to a rest Ths wll reduce the chance o dyng n a car crash Ways to ncrease the tme Seat belts Ar bags The ar bag ncreases the tme o the collson and absorbs some o the energy rom the body Suppose a png-pong ball and a bowlng ball are rollng toward you. Both hae the same momentum, and you exert the same orce to stop each. How do the tme nterals to stop them compare?. It takes less tme to stop the png-pong ball.. Both take the same tme. 3. It takes more tme to stop the png-pong ball. Suppose a png-pong ball and a bowlng ball are rollng toward you. Both hae the same momentum, and you exert the same orce to stop each. How do the tme nterals to stop them compare?. It takes less tme to stop the png-pong ball.. Both take the same tme. 3. It takes more tme to stop the png-pong ball. Please ll your answer as queston o General Purpose Answer Sheet Please ll your answer as queston o General Purpose Answer Sheet
3 Problem: Teeng O Suppose a png-pong ball and a bowlng ball are rollng toward you. Both hae the same momentum, and you exert the same orce to stop each. How do the tme nterals to stop them compare?. It takes less tme to stop the png-pong ball.. Both take the same tme. 3. It takes more tme to stop the png-pong ball. A 50-g gol ball at rest s ht by Bg Bertha club wth 500-g mass. Ater the collson, gol leaes wth elocty o 50 m/s. a) Fnd mpulse mparted to ball b) Assumng club n contact wth ball or 0.5 ms, nd aerage orce actng on gol ball Note: Because orce equals the tme rate o change o momentum, the two balls loose momentum at the same rate. I both balls ntally had the same momenta, t takes the same amount o tme to stop them. Gen: mass: m=50 g = kg elocty: =50 m/s Fnd: mpulse=? F aerage =? Problem: teeng o. Use mpulse-momentum relaton: mpulse p m m.50 kg m s kg50 m s 0. Hang ound mpulse, nd the aerage orce rom the denton o mpulse: p.50 kg m s p F t, thus F 3 t 0.50 s N Note: accordng to Newton s 3 rd law, that s also a reacton orce to club httng the ball: F t F m m MV R m MV t, or MV m MV, or CONSERVATION OF MOMENTUM Conseraton o Momentum Denton: an solated system s the one that has no external orces actng on t Momentum n an solated system n whch a collson occurs s consered (regardless o the nature o the orces between the objects) A collson may be the result o physcal contact between two objects Contact may also arse rom the electrostatc nteractons o the electrons n the surace atoms o the bodes
4 Conseraton o Momentum The prncple o conseraton o momentum states when no external orces act on a system consstng o two objects that collde wth each other, the total momentum o the system beore the collson s equal to the total momentum o the system ater the collson Conseraton o Momentum Mathematcally: m m m m Momentum s consered or the system o objects The system ncludes all the objects nteractng wth each other Assumes only nternal orces are actng durng the collson Can be generalzed to any number o objects Problem: Teeng O (cont.) Let s go back to our gol ball and club problem: Ball: p.50 kg m s, m 50 gramm Club : m 50 m s.50 kg m s, so.50 kg m s 5m s 0.5 kg actor o 0 tmes smaller Suppose a person jumps on the surace o Earth. The Earth. wll not moe at all. wll recol n the opposte drecton wth tny elocty 3. mght recol, but there s not enough normaton proded to see that could happened Please ll your answer as queston 3 o General Purpose Answer Sheet
5 Suppose a person jumps on the surace o Earth. The Earth. wll not moe at all. wll recol n the opposte drecton wth tny elocty 3. mght recol, but there s not enough normaton proded to see that could happened Suppose a person jumps on the surace o Earth. The Earth. wll not moe at all. wll recol n the opposte drecton wth tny elocty 3. mght recol, but there s not enough normaton proded to see that could happened Note: momentum s consered. Let s estmate Earth s elocty ater a jump by a 80-kg person. Suppose that ntal speed o the jump s 4 m/s, then: Please ll your answer as queston 4 o General Purpose Answer Sheet Person : p 30 kg m s Earth : V p M Earth Earth V Earth 30 kg m s, so 30 kg m s kg m s tny neglgble elocty, n opposte drecton Types o Collsons Perectly Inelastc Collsons: Momentum s consered n any collson what about knetc energy? Inelastc collsons Knetc energy s not consered KE KE lost energy Some o the knetc energy s conerted nto other types o energy such as heat, sound, work to permanently deorm an object Perectly nelastc collsons occur when the objects stck together Not all o the KE s necessarly lost When two objects stck together ater the collson, they hae undergone a perectly nelastc collson Suppose, or example, =0. Conseraton o momentum becomes m m ( m m) E.g., (000kg)(50 m s) 0 (500kg) m ) 0 ( m m m 000 kg, m 4 50 kg m s 0 m s kg 500 kg :,
6 Perectly Inelastc Collsons: What amount o KE lost durng collson? KE KE beore ater m m 6 (000 kg)(50m s).50 J ( m m ) 6 (500 kg)(0m s) J More Types o Collsons Elastc collsons both momentum and knetc energy are consered Actual collsons Most collsons all between elastc and perectly nelastc collsons KE lost J lost n heat/ glung /sound/ More About Elastc Collsons Elastc Collsons: Both momentum and knetc energy are consered Typcally hae two unknowns m m m m m m m Sole the equatons smultaneously m Usng preous example (but elastc collson s assumed) P beore KE beore m m (000 kg)(50m s) (500 kg)( 0m s) 4.00 kg m s m m J 30 J J For perectly elastc collson: 4.00 kg m s m m J m m 6.7 m s 3. m s
7 Problem Solng or One - Dmensonal Collsons Set up a coordnate axs and dene the eloctes wth respect to ths axs It s conenent to make your axs concde wth one o the ntal eloctes In your sketch, draw all the elocty ectors wth labels ncludng all the gen normaton Sketches or Collson Problems Draw beore and ater sketches Label each object nclude the drecton o elocty keep track o subscrpts Sketches or Perectly Inelastc Collsons The objects stck together Include all the elocty drectons The ater collson combnes the masses Problem Solng or One- Dmensonal Collsons, cont. Wrte the expressons or the momentum o each object beore and ater the collson Remember to nclude the approprate sgns Wrte an expresson or the total momentum beore and ater the collson Remember the momentum o the system s what s consered
8 Problem Solng or One- Dmensonal Collsons, nal I the collson s nelastc, sole the momentum equaton or the unknown Remember, KE s not consered I the collson s elastc, you can use the KE equaton to sole or two unknowns Glancng Collsons For a general collson o two objects n three-dmensonal space, the conseraton o momentum prncple mples that the total momentum o the system n each drecton s consered m m x y m x m y m x m y m x m y and Use subscrpts or dentyng the object, ntal and nal, and components Glancng Collsons The ater eloctes hae x and y components Momentum s consered n the x drecton and n the y drecton Apply separately to each drecton Problem Solng or Two- Dmensonal Collsons Set up coordnate axes and dene your eloctes wth respect to these axes It s conenent to choose the x axs to concde wth one o the ntal eloctes In your sketch, draw and label all the eloctes and nclude all the gen normaton
9 Problem Solng or Two- Dmensonal Collsons, cont Wrte expressons or the x and y components o the momentum o each object beore and ater the collson Wrte expressons or the total momentum beore and ater the collson n the x- drecton Repeat or the y-drecton Problem Solng or Two- Dmensonal Collsons, nal Sole or the unknown quanttes I the collson s nelastc, addtonal normaton s probably requred I the collson s perectly nelastc, the nal eloctes o the two objects s the same I the collson s elastc, use the KE equatons to help sole or the unknowns Rocket Propulson The operaton o a rocket depends on the law o conseraton o momentum as appled to a system, where the system s the rocket plus ts ejected uel Ths s derent than propulson on the earth where two objects exert orces on each other road on car tran on track Rocket Propulson, cont. The rocket s accelerated as a result o the thrust o the exhaust gases Ths represents the nerse o an nelastc collson Momentum s consered Knetc Energy s ncreased (at the expense o the stored energy o the rocket uel)
10 Rocket Propulson Rocket Propulson The ntal mass o the rocket s M + m M s the mass o the rocket m s the mass o the uel The ntal elocty o the rocket s The rocket s mass s M The mass o the uel, m, has been ejected The rocket s speed has ncreased to + Rocket Propulson, nal The basc equaton or rocket propulson s: M e ln M M s the ntal mass o the rocket plus uel M s the nal mass o the rocket plus any remanng uel The speed o the rocket s proportonal to the exhaust speed Thrust o a Rocket The thrust s the orce exerted on the rocket by the ejected exhaust gases The nstantaneous thrust s gen by M Ma M t e t The thrust ncreases as the exhaust speed ncreases and as the burn rate ( M/ t) ncreases
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