PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems. PATH OF A PROJECTILE 1. Lgn t a cmputer. G t the website http://www.walter-fendt.de/ph14e/prjectile.htm Change the initial height t 0. Run the simulatin. a) What is the shape f the path fllwed by the prjectile? b) Try different angles f inclinatin. Which ne gives yu the maximum distance? c) What ther angle gives yu the same distance as 30 0? As 55 0? d) What is the rule fr finding tw angles that give the same distance? VELOCITY OF A PROJECTILE 2. Click n the buttn t shw velcity. Als click n the bx fr Slw Mtin. Run the simulatin. Yu will see the verall velcity vectr as well its hrizntal and vertical cmpnents. a) Fcus n the hrizntal cmpnent nly. Run the simulatin. What des it shw fr the hrizntal velcity? Why d yu think this is happening? b) Fcus n the vertical velcity cmpnent. Run the simulatin. Describe what is happening t the vertical velcity cmpnent. What is causing this? c) Fcus n the verall velcity. Des it ever becme zer? Why r why nt? d) Lk at hw the verall velcity is riented t the path f the prjectile. What wrd d we use t describe this relatinship? 1
e) If yu increase the initial velcity, describe what happens t the distance travelled: height reached: ACCELERATION OF A PROJECTILE 3. Click n the acceleratin buttn. Run the simulatin. a) What can yu say abut the acceleratin f the prjectile? What causes it t be like this? b) Why is there n hrizntal cmpnent t the acceleratin? SOLVING PROJECTILE MOTION PROBLEMS IMPORTANT RULE: The hrizntal part f the mtin is independent frm the vertical mtin. The tw parts can be slved separately. HOWEVER, the TIME t cmplete the hrizntal mtin is the same as the TIME t cmplete the vertical mtin. CONCEPT PROBLEMS 5. A student pushes a cin ff the edge f a cunter with sme hrizntal velcity. At the instant the cin ges ver the edge, a secnd student drps a secnd cin frm cuntertp level s it falls vertically t the flr. Which cin hits the flr first? Explain yur answer. Think abut the IMPORTANT RULE abve. TRY IT. 6. A similar experiment t the ne abve is dne. A rifle is fired hrizntally ver a large flat plane. At the instant the rifle is fired, a student drps a secnd identical bullet frm the same height as the rifle. Which bullet hits the grund first? Explain. 7. A mnkey is hanging frm a tree branch. A scientist wants t sht the mnkey with a tranquilizer dart s she can study the mnkey. She knws the mnkey will be startled by the sund f the tranquilizer gun and will let g f the branch at the instant the gun is fired. Where shuld the scientist aim t hit the mnkey? (Abve it, belw it r directly at it?) Explain. T check this answer, g t yutube.cm and search fr MIT Physics Mnkey and Gun. G t full screen mde. Pause the vide just befre the prjectile is launched. Use the side f a piece f paper placed alng the edge f the gun t see where it is aimed. Remve the paper and cntinue watching the vide. Fix yur answer abve if necessary. 2
A ball is rlled at 2.2 m/s alng a cunter tp that is 1.2 m abve the flr. It rlls ver the edge f the cunter tp. a) Hw far hrizntally des it travel befre hitting the flr? b) What is the velcity f the ball when it hits the flr? SPLIT THE GIVENS INTO HORIZONTAL AND VERTICAL QUANTITIES Quantity Hrizntal (x) Vertical (y) let up be + initial velcity 2.2 m/s 0 final velcity 2.2 m/s acceleratin 0-9.8 m/s 2 displacement -1.2 m time Hrizntal: The acceleratin is 0 s the velcity is cnstant. Vertical: The acceleratin is the acceleratin due t gravity. The negative sign means dwn. The displacement is als dwn as the ball starts vertically at the cunter height and ends vertically n the flr which is lwer. Time: is the same fr bth x and y. a) Nw yu can slve the prblem as nrmal. Yu have enugh infrmatin t slve fr the remaining vertical quantities. Slve fr time: Vertical: ( ) Hrizntal: slve fr distance since v is cnstant ( ) ( ) b) The final velcity has tw cmpnents. We already knw the final hrizntal velcity (2.2 m/s). Nw find the final vertical velcity. Vertical: Cmbining: ( ) ( ) 2.2 m/s therefre the velcity f the ball n cntact with the flr is 5.3 m/s [66 0 belw hrizntal]. 4.85 m/s PRACTICE PROBLEMS D P78 Q1-2, P81 Q1-2 (Tp) Q2-8 (Bttm) 3
RELATIVE MOTION LEARNING GOALS Students will: Determine the velcity f bjects in different frames f reference WEBSITE AND TEXTBOOK RESOURCES Reading Nelsn Physics 11 Sectin 2.2 Pages 71-74 Vides Activities Train Statin What is mving? Physlets Try Relative Velcity in 1D and 2D MOTION IN 1-D Imagine yu are sitting n a train. The train is traveling at 35 km/h [W]. Yu stand up and walk frward alng the aisle f the train at 5 km/h. In this prblem there are essentially three pints f view r frames f reference: 1. the train 2. the grund 3. yu When we describe mtin, we say WHAT is mving and WITH RESPECT TO sme pint f view r frame f reference. 1. Cnsider the fllwing statements: The velcity f the train with respect t the grund is 35 km/h [W] Yur velcity with respect t the train is 5 km/h [W] Yur velcity with respect t the grund is 40 km/h [W] The velcity f the grund with respect t yu is 35 km/h [E] The velcity f the grund with respect t the train is 35 km/h [E] One f the abve statements is NOT crrect. Which ne is wrng? Explain why. 2. Yu are n the same train as abve. This time yu walk twards the back f the train at 5 km/h alng the aisle. Answer the fllwing: a) The velcity f the train with respect t the grund is b) Yur velcity with respect t the train is c) Yur velcity with respect t the grund is d) The velcity f the grund with respect t yu is e) The velcity f the grund with respect t the train is 4
NOTATION We need a shrt, mathematical way f making the same statements as abve. If sme bject A has a velcity with respect t sme ther bject r frame f reference B, we use the ntatin AV B r V AB Bth are read as velcity f A with respect t B. The first subscript is the bject, the secnd subscript is the frame f reference. We will use the secnd ntatin where bth subscripts are n the right f the variable. The rder f the subscripts is IMPORTANT. In the example we started with, the train has a velcity with respect t the grund f 35 km/h [W]. If yu are standing n the grund watching the train g by, this is what yu see. V TG = 35 km/h [W] BUT if yu are standing n the train and yu lk ut the windw, the grund appears t be ging by yu in the ppsite directin. V GT = 35 km/h [E] This gives us the rule: RULE 1: V AB = - V BA 3. Take each f the statements in questin 2 abve and rewrite them using the ntatin given abve. a) b) c) d) e) CHANGING FRAMES OF REFERENCE When yu walk alng the aisle f the train, yur velcity depends n yur chsen frame f reference. Yu culd chse the train as a frame f reference r yu culd chse the grund as the frame f reference. Yur velcity with respect t the train depends nly n yur mtin with respect t the train. Let s say yu walk backward (east) alng the aisle at 5 km/h. Yu can write V YT = 5 km/h [E]. But suppse the train is als mving alng the grund at 35 km/h [W] V TG = 35 km/h [W]. Yur mtin ver the grund is a cmbinatin f yur mtin alng the aisle f the train and the trains mtin alng the grund. Yu have prbably nticed that yu merely have t add these t get yur mtin ver the grund: V YG = V YT + V TG V YG = 5 km/h [E] + 35 km/h [W] = 30 km/h [W] NOTE: this is a vectr additin. 5
starts with A ends with C starts with A ends with C This gives us the rule: RULE 2: must be the same This rule has a pattern. The rder f the subscripts always fllws this pattern. The left side f the equatin is the velcity yu want. It has a first subscript (A) and a last subscript (C). The right side f the equatin must have the SAME first subscript (A) and the SAME last subscript (C). The inner subscripts n the right hand side must be the same. 4. Car 1 is driving at 45 km/h [W] with respect t the rad. Car 2 is driving at 52 km/h [E] with respect t the rad. What is the velcity f car 2 with respect t car 1? GIVENS velcity f car 1 with respect t the rad velcity f car 2 with respect t the rad UNKNOWN EQUATION fllw RULE 2 and write SOLUTION we need t find using RULE 1 STATEMENT Car 2 is mving at 97 km/h [E] with respect t car 1. MOTION IN 2-D Mtin in tw dimensins can be dealt with the same way as ne dimensinal mtin. BOTH RULE 1 and RULE 2 are still valid. In a typical prblem, a swimmer swims acrss a river that has a current. Whether the swimmer is in mving r still water, her velcity with respect t the water is the same. This is her swimming ability. The river current is like being n a mving train. Anther type f prblem is a plane flying while the wind is blwing. In this case, the wind is like the mving train. The plane s velcity with respect t the air is like the velcity f the swimmer with respect t the water. 6
5. A girl tries swimming acrss a river. The river is flwing at 2.3 m/s [E]. The girl swims at 1.4 m/s [N] with respect t the water. What is the girl s velcity with respect t the river bank (the shre)? write the velcity f the girl with respect t the water: write the velcity f the water with respect t the grund: write the crrect equatin using RULE 2: draw (t the side) a labelled vectr diagram t represent yur equatin: Perfrm the required peratin t get the answer: 6. A plane is flying with an airspeed f 340 km/h and a heading f west (in pilt terminlgy, airspeed and heading are the velcity with respect t the air). The plane s velcity ver the grund is 365 km/h [W16 0 S]. What is the wind velcity? write the velcity f the plane with respect t the air: write the velcity f the plane with respect t the grund: write the crrect equatin using RULE 2: draw (t the side) a labelled vectr diagram t represent yur equatin: Perfrm the required peratin t get the answer: PRACTICE PROBLEMS P74 Q2, P75Q1-10 7