ONE-DIMENSIONAL COLLISIONS

Purpose Theory ONE-DIMENSIONAL COLLISIONS a. To very the law o conservaton o lnear momentum n one-dmensonal collsons. b. To study conservaton o energy and lnear momentum n both elastc and nelastc onedmensonal collsons. c. To study mpulsve orce durng the collson. Conservaton o lnear momentum s the most mportant mplcatons o Newton s law. Lnear momentum (p) or a partcle o mass, m movng wth velocty, v, s dened as p = mv. The momentum (p) and velocty (v) are both vectors and have the same drecton. For a system o n partcles wth masses m, m, m 3,.., m n, wth respectve veloctes v, v, v 3, v n, the lnear momentum o the system s the vector sum o the ndvdual momentum,.e., p sys = p + p + + p n = m v + m v + + m n v n. () The knetc energy o a partcle o mass m and velocty v s dened as KE= ½ m v. (Note that KE s a scalar quantty). The knetc energy o the system o n partcles s gven by KE sys = KE +KE + + KE n = ½ m v + ½ m v +.+ ½ m n v n. () The conservaton o lnear momentum states that the net orce actng on a system o partcles s zero then the lnear momentum o the system s conserved (.e., p sys s constant). In solated collson problems the net lnear momentum beore and ater the collson must be same p sys beore collson = p sys ater collson (3) For the one dmensonal collson problem nvolvng two objects as shown n Fgure, we can wrte equaton (3) as p + p = p + p (4) where, reer to the ntal and nal stages. Drecton o velocty (or momentum) s taken poston when the object s movng to the rght as ndcated by arrow heads n Fgure. v v v v m m m m Beore collson Fgure. One dmensonal collson nvolvng two objects. Ater collson In our experments, we wll study the collson o two carts (cart- and cart-) movng on a horzontal track. The carts and the track are desgned or neglgble rcton. Snce there are no external orces actng on the carts durng the collson, p sys must be conserved. Addtonally, we wll keep one o Brooklyn College

the carts (cart-) at rest beore collson or all cases (.e., v = 0 hence p = 0). For ths condton, we can smply equaton (4) as m v = m v + m v (5) By measurng the masses and the veloctes beore and ater the collson, we can easly very the conservaton o lnear momentum. In ths lab, we wll nvestgate two cases. Part I: Perectly nelastc collson In the case o a perectly nelastc collson, the carts stck together ater the collson. Thus, both carts wll have the same nal veloctes, that s v = v = v. Usng ths condton n equaton (5), we get v mv. (6) ( m m ) Note: The KE sys s not conserved n a perectly nelastc collson. I the knetc energy s not conserved where does the energy go? Part II: Elastc collson By denton, n an elastc collson, the KE sys s conserved,.e. KE sys beore collson = KE sys ater collson m v m v m v (7) Solvng eqns. (5) and (7) smultaneously, we obtan or elastc collsons, v ( m m ) v and ( m m ) Part III: Impulse durng the collson v mv (8) ( m m ) ` Durng the collson each cart exerts orce on each other. Let F be the orce exerted on cart- by cart- and t s the duraton o mpact. The mpulse on the cart- s dened by the product; F. t, whch s equal to the change n momentum. Thus, F. t mv mv (9) We can wrte smlar expresson or cart- and derve equaton (4). (see queston n computaton secton) We wll attach a orce sensor on each cart to measure the orce exerted on the carts whch s reerred to as the mpulsve orce. From the graph o orce versus tme durng the collson, Impulse can be determned rom the area under the curve. Apparatus Verner rctonless carts and a track, moton detector, photogate wth tmer, orce sensors, addtonal mass, LabQuest nterace, Computer wth Logger Pro sotware, and Scale. Brooklyn College

Descrpton o Apparatus Fgure shows the expermental set up n ths lab or the case o a perectly nelastc collson. You wll use a moton detector to measure the veloctes o a cart beore and ater the collson. The moton detector should be connected to a Verner LabQuest nterace devce whch s then connected to a computer. Usng the Verner Logger Pro sotware, you can collect the data or the poston and velocty wth respect to tme. Velcro stckers on the sde o the carts make them stck together ater the collson. Added mass Velcro Moton detector Cart Cart Track Fgure : Expermental set up or perectly nelastc collson. Fgure 4 shows the expermental set up or an elastc collson. Elastc bumpers wth a Velcro stcker (or embedded magnets) are attached on the ront o the carts to make the collson elastc. Both carts wll be movng wth derent veloctes ater the collson n case o an elastc collson. So you wll need an addtonal sensor to measure the velocty o the cart-. You wll use a photogate to measure the velocty o cart- (see Fgure 3). The photogate has a tmer attached. The photogate starts the tmer when an obstacle goes past the photogate and stops the tmer when t leaves the photogate. We wll nstall a lag on cart- to start and stop the tmer. The velocty o the cart s then equal to the length o the lag dvded by the tme nterval. Photogate Tmer Fgure 3: Photogate and tmer. Moton detector Bumper or embedded magnet Flag Photogate Cart Cart Track Fgure 4: Expermental set up or elastc collson. Brooklyn College 3

Procedure. Measure the masses o the carts and addtonal loads, and record n the data sheet.. Attach a moton detector on one sde o the track. The moton detector should be about 0 cm away rom the edge o the track. The moton detector should be connected to the Verner LabQuest nterace devce whch then should be connected to the computer. Open Logger Pro n the computer. 3. Check the track s leveled. You can easly check the track s leveled or not by placng a cart on the track. I the cart stays at rest even you move the cart to another spot, t s leveled. I the track s not leveled, the cart starts movng. You should then level t by adjustng the screws under the track. Part I: Perectly nelastc collsons (For the case o perectly nelastc collson you do not need to use the photogate.) 4. Place cart- near the center o the track and the cart- at the edge o the track where the moton detector s attached. Make sure the Velcro stcker on the carts ace each other so that the carts stck together ater the collson or a perectly nelastc collson. 5. Now, get ready to make the collson and collect the data. Place you nger on the back o cart- and make sure your hand does not block the beam o the detector and that no one s standng or movng n ront o the moton detector. 6. Clck Collect on the computer. You should hear regular clckng sound rom the moton detector. Then push the cart- by your nger. DO NOT move your hand whle the carts are movng. STOP the carts ater t hts the bumper at the other edge o the track. The program wll collect the data or the poston o the cart wth respect to tme. It dsplay the data o poston and velocty n the table. It also dsplays the graphs o poston and velocty as a uncton o tme. You should see a jump n the plot at the moment the carts collde. Make sure that the velocty s approxmately constant beore and ater the collson. Repeat untl you get a nce run. Save the data and graphs or analyss. 7. Fndng the veloctes: You can get the veloctes beore and ater the collson rom both the poston and velocty versus tme graphs. a. The poston vs. tme graph should show two straght lnes o derent slopes that meet at the collson tme. Hghlght a regon o the graph usng the mouse beore the collson and clck Analyze n the menu bar and then clck Lnear Ft. A box wll appear showng the slope o the lnear tted lne. What do you get rom the slope o the poston versus tme graph? b. On the velocty vs. tme graph, hghlght the regon on the graph beore the collson tme, as done n part (a). Clck Analyze n the toolbar menu and then clck Statstcs. A box wll appear showng the average value o the velocty n the selected graph regon. Record the veloctes n Table. Check the values are close. 8. Repeat steps (7a) and (7b) by hghlghtng the regons n the graphs ater the collson tme. Record the values n Table. 9. We wll now repeat the tral wth derent masses o the carts. Add a load on cart-. Cart- s heaver than cart-. Repeat the steps 4 8 to measure the veloctes beore and ater the collson. 0. Now, remove the load rom the cart- and add onto cart-. In ths case, cart- s heaver than cart-. Repeat the measurement agan. You should nclude graphs o poston and velocty versus tme rom one o these trals n your report. Brooklyn College 4

Part II: Elastc collsons In order to realze an elastc collson, use elastc bumpers (or two magnets embedded) on the ront o the carts and place them acng each other. Put a lag on cart-. Also measure the length o the lag. You are gong to use the photogate to measure the velocty o cart- ater collson. Place the photogate about 5 cm rom the edge o the track on the sde cart- wll move.. Plug n the photogate power supply. Swtch the photogate settngs to: mode = gate, tme = 0. ms, and memory = on.. Set the approprate heght o the photogate so that cart- wth lag can pass reely through the gate and the lag also blocks the lght o the photogate (when ths happens, the red lght on the photogate lashes). The tmer s started when the ront edge o the lag passes through the photogate. The tmer s then stopped when the back edge o the lag leaves the photogate. The velocty o cart- s, thereore, the length o lag dvded by the tme nterval measured by the tmer. 3. Reset the tmer o the photogate. Repeat steps 4-0 o Part I (three derent trals). Record your measurements or cart- n Table 3. For cart-, record the tmer readngs n the table to nd the velocty o cart. The photogate must be reset beore each tral. Part III: What happens durng the collson? In the parts I and II you have nvestgated the lnear momentum o a system beore and ater the collson. What happens durng the collson? You should have observed the sudden changes n the velocty durng the collson there by changng the lnear momentum o the ndvdual carts. In ths part o the experment, you are gong to r nvestgate urther by measurng the orces the carts exert on each other durng the collson. You wll use the orce sensors to measure the orce exerted on the carts. Fgure 4 shows the schematc o the expermental set up or ths part. Cart- s loaded wth an added mass. Attach a orce sensor on each cart on the rod on the carts. Set the sensors to ±0 N range. Connect the sensor wres to the LabQuest nterace. You may remove cables o other sensors (moton sensor and photogate) rom the nterace. Open a new le n the Logger Pro program. You should see orce measured by the sensors and graphs to dsplay the orces. The orce sensor- measures the orce exerted on cart- and smlarly by sensor-. Clck at Experment on menu bar and select Set up sensors. It wll show all sensors connected to the nterace. You should see two orce sensors. Rght clck on one o the orce sensor and reverse the polarty. Why do you thnk the polarty o one orce sensor has to be reversed? Added mass Force sensor- Force sensor- Cart Cart Track Fgure 4: Expermental set up or measurng orces durng the collson. Brooklyn College 5

. Beore collectng the data, clck at Zero (0) next to Collect. Snce the collson happens n a short tme you have to ncrease samplng rate to record more data ponts. In menu bar Experment Data collecton change samplng rate to 50. Now, keep cart- around the mddle o the track at rest and cart- at some dstance away. The bumpers o the orce sensors should ace each other as shown n Fgure 4. Make sure the wres connected to the sensors do not stretch whle they move. You are now ready to see what happens durng the collson.. Clck Collect n the Logger Pro and gently push cart- towards cart-. Logger Pro wll collect the data or the orce exerted on both carts and dsplay them n the graph. How does the orce exerted on each other vary wth tme durng the collson? Compare the magntude o the orces exerted on each cart at each moment? 3. Hghlght the regon on the graph whle collson occurs or orce-. In the menu bar clck, Analyze and select Integrate. You wll see the area under the plot lled and a small box wth the value o area whch s essentally Impulse. Repeat t or orce-. Compare the value o Impulses. 4. Now, nterchange the postons o the carts and repeat the prevous steps. 5. What do you expect about the orce exerted on each cart cart- s pushng cart- and both movng together? Test your predcton. Computaton From the data you collected n Tables and 3 calculate the theoretcal nal veloctes usng Equatons (6) and (8) and compare wth the expermental values. Calculate the total lnear momentum rom the expermental results beore and ater the collson or each tral. Do your measurements ndcate conservaton o lnear momentum? I no, explan the possble sources o error. Calculate the total knetc energy beore and ater the collson or each tral n Tables and 4. Do your measurements ndcate conservaton o knetc energy? Explan the possble sources o error. Explan your observaton and the concluson rom Part III. Based on your concluson rom the Part III o ths lab, derve Equaton 4. Questons. What are the derences n your expermental results between the perectly elastc and nelastc collsons?. I the masses o the carts are equal n the elastc collson, what should happen to the carts ater the KE collson? What happened n your experment? Derve the expresson or rato,, and compare KE the theoretcal values wth your expermental result. 3. When a truck and a car are travelng wth same speed make a head-on collson, s the orce exerted by the car on the truck equal, greater or lesser than that exerted by the truck on the car? 4. Explan, usng lnear momentum, why a rle must recol n the opposte drecton o the bullet when red? Brooklyn College 6

Date experment perormed: Name o the group members: Perectly nelastc collson Data Sheet Mass o empty cart- = ; Mass o empty cart- = ; Mass o the load = Table : Velocty and Momentum (Note v or cart- s zero) Tral # 3 Beore collson Ater collson (Carts stocked together) Cart- Cart- Cart- Expt. v Theory m m v p sys (kg.m/s) Graph- (x vs t) Graph- (v vs t) Average v p sys (kg.m/s) Change n p sys (%) Are the theoretcally predcted and expermentally observed values or the nal velocty the same? Is the lnear momentum conserved or all trals? Table : Knetc energy Tral # Beore collson Cart- Cart- Cart- Cart- Stocked carts m m v v 0 0 3 0 KE sys (J) Average Expt. v Ater collson KE sys (J) Change n KE sys (%) Is the knetc energy conserved n all trals? Brooklyn College 7

Elastc collson Mass o empty cart- = ; Mass o empty cart- = ; Mass o the load = Length o the lag, L = Table 3: Velocty and Momentum (Note v or cart- s zero) Tral # 3 Beore collson Ater collson Cart- Cart- Cart- Cart- Cart- m m v p sys (kg.m/s) graph- (x vs t) Expt. v graph- (v vs t) average Theory v t (s) v = L/ t p sy (kg.m/s) Change n p sys (%) Are the theoretcally predcted and expermentally observed values o the nal velocty the same? Is the lnear momentum conserved or all trals? Table 4: Knetc energy Tral # Beore collson Ater collson Cart- Cart- Cart- Cart- Cart- Cart- m m v v KE sys (J) Expt. v Expt. v KE sys (J) Change n KE sys (%) 0 0 3 0 Is the knetc energy conserved n all trals? Brooklyn College 8