AP Phsics C Unit 2 2.1 Nme Vectos Vectos e used to epesent quntities tht e chcteized b mgnitude ( numeicl vlue with ppopite units) nd diection. The usul emple is the displcement vecto. A quntit with onl mgnitude nd no diection is clled scl. Vectos e smbolized in the tet with supeposed ow (eg. ). The scl mgnitude of the vecto is smbolized using bsolute vlue signs (eg. ) o simpl b the lette itself (eg. ). We gee fo the time being tht vecto cn be moved s long s its length nd diection e peseved. To dd two vectos, plce the til of the second t the tip of the fist. The vecto sum, o esultnt, is the vecto fom the til of the fist to the tip of the second. The set of vectos is sid to be closed unde vecto ddition; tht is, the sum of two vectos is gin vecto. In the spce below, constuct the vectos + b nd b + to show tht vecto ddition is commuttive ( + b = b + ). b + b b + In the spce below, constuct the vectos + ( b + c) nd ( + b) + c to show tht vecto ddition is ssocitive ( + ( b + c) = ( + b) + c ). b c + ( b + c) ( + b) + c The identit element fo vecto ddition is tht vecto which, when dded to n bit vecto, ields the vecto itself. The identit element fo vecto ddition is clled the zeo vecto. It hs no length nd n undefined diection. Fo given vecto, the dditive invese is tht vecto which, when dded to, ields the identit element (the zeo vecto). We smbolize this vecto s. Descibe the length nd diection of eltive to : (FYI: A set of mthemticl objects (eg., vectos) togethe with n opetion (eg., vecto ddition), tht obes the ules of closue, ssocitivit, identit nd invese is clled goup. A goup tht lso obes commuttivit is clled n Abelin goup fte the mthemticin Niels Abel. Goups e fundmentl to the stud of element pticles.) Cn two vectos whose mgnitudes e diffeent be dded to give zeo esultnt? Wh o wh not? Cn thee vectos whose mgnitudes e diffeent be dded to give zeo esultnt? Wh o wh not? Cn the sum of the mgnitudes of two vectos eve be equl to the mgnitude of the sum of those vectos? Eplin.
Wht e the popeties of two vectos nd b such tht A) + b = c nd + b = c? 2.2 B) + b = c nd 2 + b 2 = c 2? Components nd Unit Vectos A given vecto cn be epessed s the sum of two othe vectos in n infinite viet of ws. Given coodinte sstem, it is useful to epess vectos in tems of the sum of two vectos, ech of which points pllel to n is. These e clled the components of the oiginl vecto. Fo vecto in the - plne, its components e clled nd. Dw these vectos in the spce to the ight. Epess the following in tems of, the length of, nd : = = nd in tems of nd : = Vecto equtions in n dimensions e shothnd fo n scl equtions t once. Fo emple, the vecto eqution = b + c in thee dimensions is equivlent to the thee scl equtions = b + c, = b + c nd z = b z + c z. We define unit vectos in the,, nd z diections s vectos with mgnitude 1, nd cll them î, ĵ, nd ˆk espectivel. Given n bit vecto, epess it in tems of the mgnitudes of its components nd the unit vectos: = Cn vecto hve zeo mgnitude if one of its components is non-zeo? Eplin. Multipliction of Vectos Thee e thee kinds of vecto multipliction: Scl multipliction, dot poduct, nd coss poduct. 1. Scl multipliction is the multipliction of vecto b scl. The esult is vecto colline with the oiginl vecto. If the scl is pue numbe, then the esult is vecto of the sme kind s the oiginl, but if the scl hs units, then the esult is vecto of diffeent tpe. Given the vecto v below, with mgnitude v = 2 m/s, dw the vectos tht esult fom the scl multipliction b the given scl, nd indicte the mgnitude of ech. v scl: 2; mgnitude: scl: 3 s; mgnitude: Note tht the vecto ou dw in the second bo does not hve to be thee times s long s v. The scles of vectos of diffeent tpes e independent of ech othe.
2. The dot poduct of two vectos nd b is smbolized b i b. It is the scl bcos, whee is the ngle between the two vectos nd b when plced til-to-til. Is the set of vectos closed unde this opetion? Wh o wh not? b 2.3 Is the dot poduct commuttive? Wh o wh not? Is the dot poduct ssocitive? Wh o wh not? Conside the following sttement: i b = b = b b whee, fo emple, b mens the length of the component of b in the diection of. In the spce to the ight, show how ou cn find b nd b. Show wh the sttement follows fom the definition of dot poduct. b b : b : b Conside ving the ngle between two vectos nd b. Wht is the ngle tht mkes i b the getest? = Then i b = Wht ngle mkes i b the lest? = Then i b = If i b = i c, does it follow tht b = c? Wh o wh not? Eplin how one could use the dot poduct to find the ngle between two vectos. Conside the tble to the ight, listing ll possible dot poducts of the unit vectos. Of the 9 dot poducts, how mn e zeo? Fill these in nd eplin wh. î i î = ĵ i ĵ = ˆk i ˆk = î i ĵ = î i ˆk = ĵ i ˆk = Wht is the vlue of the emining dot poducts in the tble? Fill these in nd eplin wh. (Ae these vlues vectos o scls?) ĵ i î = ˆk i î = ˆk i ĵ =
The dot poduct of nd b cn be epessed in tems of the unit vectos s i b = ( î + ĵ + z ˆk )i( b î + b ĵ + b z ˆk ). The distibutive lw holds fo dot poducts, giving totl of 9 tems when the poduct is witten out. Of the 9 tems, how mn e non-zeo? Eplin. 2.4 Epess the dot poduct of nd b in tems of thei si components: i b = 3. The coss poduct of two vectos nd b is vecto smbolized b b. Its mgnitude is bsin, whee is the ngle b between the two vectos nd b when plced til-to-til. The diection of b is pependicul to both nd b, nd is found b the ight-hnd ule: Cul the finges of ou ight hnd fom to b. You thumb points in the diection of b. Is the set of vectos closed unde this opetion? Wh o wh not? (Hint: Wht if the vectos e not dimensionless?) nd b Plne contining b Is the coss poduct commuttive? Wh o wh not? Is the coss poduct ssocitive? Wh o wh not? (Hint: Conside thee diffeent vectos in the sme plne.) Conside the two vectos nd b shown til-to-til to the ight with ngle between them. We complete the pllelogm s shown in the digm. Wht does the mgnitude of the vecto b hve to do with this pllelogm? Eplin. b Conside ving the ngle between two vectos nd b. Wht is the ngle tht mkes b the getest? = Then b = Wht ngle mkes b the lest? = Then b = Conside the tble to the ight, listing ll possible coss poducts of the unit vectos. Of the 9 coss poducts, how mn e zeo? Fill these in nd eplin wh. î î = ĵ ĵ = ˆk ˆk = î ĵ = î ˆk = ĵ ˆk = Wht is the mgnitude of the emining coss poducts? Eplin. ĵ î = ˆk î = ˆk ĵ = Use the ight-hnd ule to fill in the emining coss poducts in the tble, using unit vecto nottion.
The coss poduct of nd b cn be epessed in tems of the unit vectos s b = î + ĵ + z ˆk ( ) ( b î + b ĵ + b z ˆk ). The distibutive lw holds fo coss poducts, giving totl of 9 tems when the poduct is witten out. Of the 9 tems, how mn e non-zeo? Eplin. 2.5 Epess the coss poduct of nd b in tems of thei si components nd the unit vectos: b = Emples 1. Given the vectos nd b shown to the ight, epess the following in tems of the unit vectos (whee ppopite). Assume tht the positive z diection is out of the pge. A) the vecto : B) the vecto b : b 60 30 = 8 b = 3 C) the poduct i b : D) the poduct b : 2. Given the vectos nd b shown to the ight, epess the following in tems of the unit vectos (whee ppopite). Assume tht the positive z diection is out of the pge. A) + b : 37 B) C) b : b : = 4 b = 10 b D) i b : E) b : F) b : 3. Simplif the following poducts: A) ĵ i î = B) î ˆk = C) ĵ i( î ˆk )=
4. Given = 3î + 2 ĵ ˆk nd b = 2î + 4 ˆk, find the mgnitude of the vecto poduct b in bit units. 2.6 5. Detemine the vlue of n so tht = 2î + nĵ + ˆk is pependicul to b = 4î 2 ĵ 2 ˆk. 6. Find the ngle between the vectos î + ĵ + ˆk nd î ĵ ˆk. Pticle Kinemtics in 2 nd 3 Dimensions In the one-dimensionl cse, we took position () nd time (t) s fundmentl quntities, defined velocit (v) nd cceletion () in tems of time deivtives, nd woked out the kinemtics b integtion. In multiple dimensions, the diffeence is tht position is now vecto ( ) with components (eg.,, nd z in thee dimensions). We define the velocit vecto s the deivtive of the position vecto: v = d dt As lws, this vecto eqution stnds fo thee scl equtions t once: v =, v =, nd v z =, z If hs constnt length, is it possible fo d dt to be nonzeo? Eplin. z We define the cceletion vecto s the deivtive of the velocit vecto: = If cceletion is constnt (both in mgnitude nd in diection), then the kinemtic equtions in vecto fom cn be deived in mnne simil to the one-dimensionl cse. Wite the definition of cceletion s diffeentil eqution: Choose ppopite limits nd wite the integl (limits cn be vectos o scls, s ppopite):
Integte to get the eqution tht coesponds to v = v o + t. 2.7 Wite this s diffeentil eqution in : Then integte gin to get the eqution tht coesponds to = o + v o t + 1 2 t2. Now combine these to get the vecto equtions tht coespond to the othe two. Fist solve the fist eqution fo nd plug into the second to get the eqution tht coesponds to = o + v + v o 2 t : Then tke the dot poduct of the fist eqution with itself, nd use the distibutive lw nd the second eqution to put it in the ( ) : fom tht coesponds to v 2 = v o 2 + 2 o Recll fom Unit 1 (pge 1.2) tht we esticted ou stud of kinemtics to igid, non-otting objects in one dimension. We now conside emples of two-dimensionl motion; the etension to thee (o moe) dimensions is stightfowd once we undestnd vecto nottion.
Pojectile Motion Neglecting i esistnce, pticle pojected ne the sufce of the eth undegoes two simultneous motions: In the veticl plne (the -diection), it undegoes motion with constnt cceletion ( = g, ). Hoizontll (the - 2.8 diection) it moves with constnt velocit ( = 0 ), so we onl need to conside thee hoizontl vibles:, v, nd t. The fou kinemtic equtions simplif to one: = v t. Pojectile motion occus duing time intevl, s fom t = 0 to t = t. The vibles tht descibe the motion e given in the following tble: Veticl Hoizontl Vible Mening Vible Mening The net veticl displcement fom the beginning to the end of the intevl Alws g, the cceletion due to gvit, diected The net hoizontl displcement fom the beginning to the end of the intevl. v o The veticl component of the velocit t the beginning of the intevl (t t = 0). v The constnt hoizontl velocit thoughout the intevl v t The veticl component of the velocit t the end of the intevl (t t = t). The elpsed time fom the beginning to the end of the intevl t The elpsed time fom the beginning to the end of the intevl v o v o v t = 0 t = t v = g v v Solving pojectile motion poblems mounts to setting up tbles fo veticl nd hoizontl motions like the one to the ight, nd noting tht, since the motion is simultneous, the t in both columns is the sme. vet = = t = v o = v = hoiz = v = t =
Emples 1. A pojectile is lunched fom the oigin of the - plne with n initil speed v o t n ngle fom the hoizontl. Show tht is qudtic function of, so tht the tjecto is pbol. v o 2.9 2. A golf bll is t distnce d fom the bse of tee of height H s shown to the ight. Show tht in ode to chip the bll so tht it just cles the tee t the pek of its flight, the golfe must choose club whose loft ims the bll t point diectl bove the tee, twice the height of the tee. (ie., show tht tn = 2H d ) H H d
3. A tennis ple hlf-volles bll fom point 6 m fom the net, nd it just cles the net (which is 1 m high t tht point) t the pek of its flight. (A hlf-volle occus when the bll is stuck just fte it hits the gound.) Detemine the initil velocit nd the initil ngle t which the bll ws hlf-volleed. 6 m 1 m 2.10 4. A cnnon shoots pojectile on level gound t n initil speed v o nd ngle s shown to the ight. v o A) Detemine the nge R of the pojectile. R B) Show tht the mimum nge R m is chieved when = 45.
Fo pojectile shot t 45, detemine the following in tems of R m nd constnts: C) the mimum height eched 2.11 D) the time of flight Unifom Cicul Motion A pticle moving in cicul pth of dius t constnt speed v is in unifom cicul motion. Note tht the speed v is constnt but the velocit v is not, since its diection (tngent to the cicle t ll times) is not. If the time it tkes fo the pticle to go completel ound the cicle is T (clled the peiod), then v the speed is: v = Conside time t, shote thn T. Let s epesent the distnce long the c tht the pticle tvels in time t. Epess s in tems of v nd t: cicle s cente. Epess s in tems of nd : Let be the ngle subtended b s t the Combine these two equtions to s eliminte s: Solve this eqution fo : Tke the deivtive with espect to t: d dt = This te of chnge of ngle is clled the ngul velocit.
To completel descibe the kinemtics of unifom cicul motion, we need epessions fo the position, velocit nd cceletion vectos of the pticle t n time. Using pol coodintes mkes these epessions simple, but equies the definition of pol unit vectos. Conside pticle moving t constnt speed v in cicle of dius. Its pol coodintes e (, ). We define the pol unit vectos s vectos of length 1; û points in the sme diection s û 2.12 û (dill out fom the cente of the cicle) nd û points pependicul to û, tngent to the cicle in the sme diection s inceses (ie., counteclockwise). Epess the position vecto in tems of pol unit vectos nd the ppopite scl: Epess the velocit vecto in tems of pol unit vectos nd the ppopite scl: = v = Unlike the Ctesin unit vectos î nd ĵ, which e constnt, the pol unit vectos chnge diection, so the e not. The theefoe hve nonzeo deivtives. Use the digm to the ight to epess û nd û in tems of the Ctesin unit vectos nd the ngle : û ĵ û î û = û = Find the time deivtive of û. You will need to use the chin ule since is function of time: dû dt = Use the definition of ngul velocit to epess this deivtive in tems of v,, nd pol unit vectos: dû dt = We e now ed to epess the cceletion vecto in tems of pol unit vectos nd the ppopite scls. Recll tht v is constnt, but v is not: = d v dt = Wht does this epession s bout the mgnitude nd the diection of the cceletion in unifom cicul motion? The Genel Cse of Two-Dimensionl Motion The genel cse of n motion whtsoeve in two dimensions cn be woked out bsed on the epessions bove fo unifom cicul motion. We hve esticted the pticle to constnt speed v nd constnt dius. If we el these two estictions, we hve pticle tht cn chnge its speed, nd chnge the dius of cuvtue of its pth; this is genel enough to descibe n two-dimensionl motion. Epess the cceletion vecto if the speed v isn t constnt. You will need to use the poduct ule: = d v dt = This epession beks the cceletion vecto into two components, one centipetl (cente-seeking) nd one tngentil. Recll tht cceletion mens te of chnge of velocit, nd tht the velocit vecto cn chnge in two ws: mgnitude nd diection. The centipetl component doesn't chnge the mgnitude (the speed), onl the diection. The tngentil component doesn't chnge the diection, onl the speed.
2.13 In the genel cse, we el the estiction tht the pticle moves in cicle of constnt dius. We define the instntneous dius of cuvtue of pth s the dius of the cicle tht mtches the pth ove diffeentill shot c. The in the epession on the pevious pge is then intepeted s the instntneous dius of cuvtue of the pticle s pth. Emples 1. An object is moving long n ovl tck t constnt speed s shown to the ight. The dots epesent successive positions t equl time intevls. The velocit vecto fo the object is dwn t point E. A) On the digm, dw velocit vectos t points A - D consistent with the given vecto v E. B C D B) Fo the time intevl fom B to C, constuct the vecto epesenting the chnge in velocit Δ v BC C) Imgine the point C moving close to point B. As it does so, how does the diection of Δ v BC chnge? A v E E D) In the limit, s point C ppoches point B, the Δ v BC vecto becomes the d v B vecto. Wht is the diection of this vecto comped to the v B vecto? E) On the digm, dw vecto epesenting the cceletion of the object t point B. Eplin how ou know its diection. F) On the digm, constuct the vecto epesenting Δ v AB. How does the length of Δ v AB compe to the length of Δ v BC? G) On the digm, dw vecto epesenting the cceletion t A, consistent with ou B vecto. H) Genelize the discussion bove to mke sttement bout the behvio of the cceletion vecto of pticle moving t constnt speed long cuve with ving dius of cuvtue.
2. An object is moving long n ovl tck t incesing speed s shown to the ight. The dots epesent successive positions t equl time intevls. The velocit vecto fo the object is dwn t point A. A) Dw velocit vectos t points B - D consistent with the given vecto v A. B) Fo the time intevl fom B to C, constuct the vecto epesenting the chnge in velocit Δ v BC C) Imgine the point C moving close to point B. As it does so, how does the diection of Δ v BC chnge? v A A B C D 2.14 D) In the limit, s point C ppoches point B, the Δ v BC vecto becomes the d v B vecto. Wht is the diection of this vecto comped to the v B vecto? How is it diffeent fom the pevious emple? E) On the digm, dw vecto epesenting the cceletion of the object t point B. F) Mke genel sttement bout pticle moving t ving speed long given pth, comped to pticle moving with constnt speed long the sme pth. 3. Below is cuve epesenting the tjecto of pticle in two dimensions. The dots epesent successive positions of the pticle t equl time intevls. At the letteed points A - G, dw esonble vectos epesenting the centipetl nd tngentil components of the cceletion. If eithe is bout zeo, stte tht eplicitl. F B C D E G A
2.15 4. Below is pbol epesenting the tjecto of pojectile. Ech dot epesents successive positions t equl time intevls. The cceletion t eve point on the pbol hs the sme mgnitude nd diection; it is g, the cceletion due to gvit, nd the digm shows these vectos. A) Dw vectos t ech point epesenting the centipetl nd tngentil components of the cceletion ( c nd t ), consistent with the given vectos. C D E B F A G g B) Using the tems slowing nd speeding up, descibe the motion of the pojectile fom A to G in tems of the tngentil component of the cceletion. C) Using the tem dius of cuvtue, descibe the motion of the pojectile fom A to G in tems of the centipetl component of the cceletion.
2.16 Instnt 1 Instnt 2 Instnt 3 Instnt 4 Instnt 5 Acceletion Velocit Desciption Constnt speed Constnt speed Constnt speed Constnt speed Constnt speed 5. Ech digm bove shows cceletion nd velocit vectos fo n object t diffeent instnts in time. Fo ech instnt, check the bo coesponding to the coect desciption. Then wite genel sttement below using the concept of dot poduct tht summizes the esoning ou used to check the boes. 6. An object moves clockwise long the tjecto shown below, looking fom the top. The cceletion vies, but is lws diected t point K. B C D A K E G F A) Dw ows on the digm t points A G to indicte the diection of the cceletion t ech point. B) Fo ech point, check whethe the object is speeding up, slowing, o neithe. A B C D E F G Neithe Neithe Neithe Neithe Neithe Neithe Neithe
2.17 7. An object moves clockwise once ound the tck shown to the ight, looking fom the top. Stting fom est t point A, it speeds up t constnt te until just pst point C, nd b the time it eches point D it is tveling t constnt speed. It then tvels t constnt speed the est of the w ound the tck. Stight line A B C F D G E A) On the digm, dw cceletion vectos t ech point; if the cceletion is zeo, stte tht eplicitl. B) How does the mgnitude of the cceletion t E compe to tht t G? Eplin. 8. A pticle tvels with constnt speed counteclockwise in cicle of dius 3 m s shown to the ight, completing one evolution in 20 s. The bottom of the cicle is chosen s the oigin of the coodinte sstem, whee the pticle stts t t = 0. Detemine the following in tems of the unit vectos î nd ĵ : A) The position vecto t t = 5 s B) The position vecto t t = 7.5 s C) The pticle s displcement vecto duing the intevl fom t = 5 s to t = 10 s D) The pticle s vege velocit vecto fo the intevl fom t = 5 s to t = 10 s E) The pticle s velocit vecto t t = 5 s F) The pticle s velocit vecto t t = 10 s G) The pticle s cceletion vecto t t = 5 s H) The pticle s cceletion vecto t t = 10 s