TIInterActive FileP.Il moto parabolico e la parabola nel piano cartesiano (si prevedono circa 15 ore di lavoro in classe) Prerequisiti: funzioni lineari, successioni lineari; concetto di funzione; differenze finite; polinomi e operazioni con essi; il moto rettilineo uniforme e la retta nel piano cartesiano. Premessa Proseguiamo con il problema dei moti e con l'introduzione di alcune particolari curve con i metodi della geometria analitica, occupandoci del moto di un corpo che segue una traiettoria parabolica e della parabola nel piano cartesiano. Scheda 1. Moto rettilineo non uniforme (Si prevedono circa 2 ore di lavoro in classe) Nella precedente lezione abbiamo imparato che: a) quando si dice che un corpo si muove di moto rettilineo, vuol dire che la sua traiettoria, ossia l'insieme dei punti che effettivamente percorre, una retta; b) un corpo, sul quale non agiscono forze, si muove di moto rettilineo se e solo se la sua velocit costante, ossia se e solo se il suo moto rettilineo uniforme; c) un corpo, vincolato a un binario rettilineo, potrebbe muoversi a velocit non costante; d) se un corpo si muove di moto rettilineo uniforme, non solo la sua traiettoria, ma anche le funzioni orarie che individuano come variano le sue coordinate rispetto al tempo, sono descritte da funzioni lineari. Occupiamoci ora di un corpo vincolato a muoversi su una retta, ma la cui velocit varia; si tratta quindi di moto rettilineo, ma non uniforme. Se il moto avviene lungo una retta, la variazione della posizione rispetto al tempo pu essere descritta da un'unica coordinata, diciamo x. Cerchiamo quindi la funzione oraria che descrive la variazione di x rispetto al tempo. Se la velocit varia, significa che presente un'accelerazione. Consideriamo qui il caso in cui l'accelerazione a sia costante, ossia non vari nel tempo. La sua funzione oraria quindi: a = costante Chiamiamo tale moto moto rettilineo uniformemente accelerato. Chiamiamo a l'accelerazione costante e chiediamoci se la sua conoscenza consente di determinare come varia la velocit nel tempo. L'accelerazione definita come una variazione di velocit nel tempo; quindi abbiamo che  (per h che tende a 0) Possiamo quindi pensare all'accelerazione come alla derivata della funzione v=v(t), ossia della funzione che vogliamo trovare. Siamo quindi di fronte al problema gi affrontato in cui conosciamo una funzione f e vogliamo trovare quella funzione F tale che F' = f. Nel caso particolare vi si chiede di determinare una funzione la cui derivata una costante a Domanda 1. Provate a scrivere una funzione v= v(t) la cui derivata sia uguale alla costante a La nostra risposta Domanda 2 Dovreste aver trovato che v = v(t) una funzione lineare del tempo avente come pendenza proprio a. Dovreste quindi aver trovato una funzione oraria di variazione della velocit di questo tipo: v= v0 + a*t Vi siete ricordati di mettere nella funzione oraria una costante come v0 che indichi la velocit all'istante iniziale? La nostra risposta ed esplicitazione di eventuali parti non comprese Domanda 3. Abbiamo visto che, nel moto uniformemente accelerato, la velocit varia linearmente con una legge del tipo v= v0 + a*t Chiediamoci ora se la conoscenza di tale legge consente di determinare come varia la posizione del corpo nel tempo. La velocit istantanea definita come una variazione di posizione nel tempo quando l'intervallo di tempo considerato tende a 0; quindi abbiamo che  (per h che tende a 0) Possiamo quindi pensare alla velocot istantanea come alla derivata della funzione x=x(t), ossia della funzione che vogliamo trovare. Siamo quindi di fronte al problema gi affrontato in cui conosciamo una funzione f e vogliamo trovare quella funzione F tale che F' = f. Nel caso particolare vi si chiede di determinare una funzione la cui derivata una funzione lineare del tipo v0 + a*t Domanda 4. Provate a scrivere una funzione x= x(t) la cui derivata sia uguale alla funzione lineare v0 + a*t La nostra risposta Domanda 5 Dovreste aver trovato che x = x(t) una funzione che varia quadraticamente nel tempo; la funzione oraria x=x(t) del seguente tipo: x= x0 + v0*t +0.5 * a* t^2 Vi siete ricordati di mettere nella funzione oraria una costante come x0 che indichi la posizione all'istante iniziale? La nostra risposta ed esplicitazione di eventuali parti non comprese Domanda 6 Considerate il moto di una pallina lanciata verso l'alto (trascurate l'attrito dell'aria): si tratta di un moto rettilineo uniformmente accelerato (perch?). Determinate le funzioni orarie della velocit e della posizione rispetto al tempo. La nostra risposta ed esplicitazione di eventuali parti non comprese Domanda 7 Considerate una pallina che viene lanciata verso l'alto, dall'altezza (rispetto terra) di 1 m, con una velocit di 3 m/s (trascurate l'attriti dell'aria). Dopo quanto tempo ricade a terra? Qual la massima altezza raggiunta? A che altezza si trova dopo 1,3 secondi? Dopo quanto tempo ripassa per il punto da cui stata lanciata? La nostra risposta Le funzioni quadratiche che esprimono la variazione della posizione nel tempo di un corpo che si muove di moto rettilineo uniformmente accelerato hanno come grafici curve dette parabole, che hai studiato nella lezione relativa alle funzioni quadratiche. Nel prosieguo del lavoro approfondiremo lo studio delle parabole dal punto di vista della geometria analitica. Precisiamo che, in generale, definiamo: a) moto rettilineo uniforme, un moto in cui costante il vettore velocit; b) moto uniformemente accelerato, un moto in cui costante il vettore accelerazione. Scheda 2. Il moto parabolico e l'equazione della parabola (Si prevedono circa 7 ore di lavoro in classe) Abbiamo appena visto che l'equazione oraria di un moto uniformemente accelerato una funzione quadratica del tempo. In precedenza abbiamo per considerato il moto uniformemente accelerato di un corpo vincolato a muoversi su un binario rettilineo, quindi in una dimensione. Occupiamoci ora del moto di un punto in un piano, per la precisione del moto di un corpo puntiforme che sottoposto alla forza di gravit e che viene lanciato con una velocit v0 che forma un angolo a con la direzione dell'accelerazione di gravit. Il vettore accelerazione di gravit si pu supporre costante (il cui modulo indichiamo con g), quindi si tratta di un moto uniformemente accelerato; d'altra parte non si tratta pi di un modo rettilineo, a meno che la velocit iniziale non formi un angolo nullo o di 180 con la direzione dell'accelerazione di gravit. In tutti gli altri casi abbiamo un moto che avviene sul piano e che quindi pu essere descritto dalle equazioni orarie dell'ascissa e dell'ordinata del punto (qui e nel seguito supponiamo trascurabile l'attrito dell'aria). Attivit 1. (attivit individuale) Prima di proseguire nella lettura prova a descrivere le caratteristiche del moto di un corpo che sottoposto alla forza di gravit e che viene lanciato con una velocit iniziale che forma un angolo a con la direzione dell'accelerazione di gravit. La mia risposta Attivit 2. (in piccoli gruppi) Confrontate, nel piccolo gruppo di lavoro, le risposte prima fornite Eventuali significative differenze fra le risposte che abbiamo fornito individualmente Ci proponiamo ora di descrivere con precisione il moto di un proiettile. Innanzitutto scriviamo i dati, ossia l'accelerazione di gravit g costante e diretta verso il basso e la velocit iniziale v0, che forma un angolo a con la direzione dell'accelerazione di gravit. In secondo luogo scegliamo un sistema di riferimento opportuno, la cui origine, per esempio, coincida con il punto in cui si trova il proiettile quando viene lanciato, il cui asse verticale abbia la stessa direzione dell'accelerazione di gravit, ma abbia senso opposto. Siamo ora pronti a scrivere le funzioni orarie dell'ascissa x e dell'ordinata y del punto (stiamo implicitamente applicando il principo di scomposizione dei moti che afferma che il moto di un punto nel piano pu essere descritto precisando come variano rispetto al tempo le coordinate di quel punto). Iniziamo dall'ascissa: l'asse delle x perpendicolare alla direzione dell'accelerazione di gravit, quindi non vi sono componenti dell'accelerazione lungo x. Per tale motivo il moto dell'ascissa del punto rettilineo uniforme, ossia l'ascissa varia linearmente nel tempo. Per come abbiamo scelto l'origine del sistema di riferimento abbiamo che: x= v0x * t Domanda 1. Per quale motivo abbiamo scritto v0x e non semplicemente v0 ? La nostra risposta (precisare eventuali divergenze fra i componenti del gruppo) Vediamo ora come varia l'ordinata rispetto al tempo. In tal caso abbiamo un moto uniformemente accelerato con accelerazione g. Per come abbiamo scelto il sistema di riferimento, abbiamo: y = v0y * t - 0.5 *g* t^2 Domanda 2. Per quale motivo abbiamo scritto v0y e non semplicemente v0 ? Perch abbiamo scritto - 0.5 *g* t^2 e non +0.5 *g * t^2? La nostra risposta (precisare eventuali divergenze fra i componenti del gruppo) Siamo ora nella situazione gi affrontata nel moto rettilineo unforme, quando ci siamo chiesti come risalire alla traiettoria di un punto per cui si conoscevano le funzioni orarie della sua ascissa e della sua ordinata. Abbiamo visto che il problema si risolve ricavando il tempo da una delle due equazioni e sostituendo l'espressione cos trovata nell'altra. Quindi, dalla conoscenza che le coordinate di P sono (v0x * t; v0y * t - 0.5 *g* t^2 ) possiamo ricavare la traiettoria di P ricavando t = x/ v0x e sostituendo il valore di t cos trovato nell'espressione dell'ordinata ottenendo: y = v0y * t - 0.5 *g* t^2  Come potete vedere l'equazione della traiettoria quella di una funzione quadratica il cui grafico una parabola. Se aprite il file di CabriIIplus o il corrispondente file di cabrijava, potrete fare diverse simulazioni del lancio di un proiettile variando la velocit iniziale sia in intensit che in direzione (per variare la direzione della velocit iniziale potete muovere il punto P sulla circonferenza; per variare l'intensit della velocit iniziale variare il raggio della circonferenza). Il file stato costruito in questo modo: a) si tracciata una retta su cui stato messo un punto la cui ascissa (t) descrive il variare del tempo; b) si tracciata una circonferenza il cui raggio rappresenta la velovit iniziale e le cui proiezioni su direzioni parallele agli assi cartesiani rappresentano v0x e v0y); c) si sono scritte, mediante lo strumento calcolatrice, le due funzioni orarie lungo x e lungo y; d) si riportato sull'asse della x il numero che esprime i valori assunti, al variare del tempo, dall'ascissa del punto P; una cosa analoga stata fatta sull'asse delle y. e) A questo punto si individuato il punto di coordinate (x; y) e si chiesto il luogo di P al variare di t. Il segmento in tratto spesso nero rappresenta l'altezza di un muro che dovreste superare fornendo, per una fissata velocit, una particolare angolazione di tiro. Attivit 3. (in piccoli gruppi) Esplorate la situazione e descrivete quello che osservate; cercate anche di prevedere quali possono essere gli effetti dei cambiamenti su direzione e intensit della velocit iniziale. Cercate anche di individuare, al variare dei parametri del sistema (altezza del muro e intensit della velocit iniziale) per quali angoli di tiro il proiettile riuscir a superare il muro e per quali angoli andr a sbattere sul muro. Guardate anche come varia la gittata (ossia la lunghezza del cammino percorso in orizzontale prima di cadere a terra) al variare dell'angolo di tiro. X6P7P91PjPkP&P'P/P1P&PE@PF@P5@P6E@P{@P}@P@PI@PL@PMT@P@P@P@Py@P @P @P P SPX PY P\ Po P P P @P @P @P @P @P @P @@P @P e@PW @PX @P_ GP P .P HP @P* P5 p@P @P @P @P @PV@PW@P_ PiJP@P@P@PPP%P  PP@P/@P0@P3@P4@P5@P6k@P@P@P@P@PIPP/P1HPy@P PP@PFP@P PPN@P$P8<@Pt;P@P@P@P@P@PR@P P@P@P3@P-P= @PG PSPgG@PVP@P@P@P@P@P!@P!@P! 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0 0 0 ffff0000 0 1 0 80170004 0 0 0 0 10001 0 0 9280000 2e80 6900 5300 0 ffff00 1000000 0 1000000 2e800b00 69000000 53000000 0 ff000000 ff 100 0 9a000200 2d80 6900 5300 0 ffff00 1000000 0 1000000 2d800b00 69000000 53000000 0 ff000000 ff 100 0 10000100 2d80 6900 5300 0 1ffff00 1000000 0 32000000 0 0 130000 0 1801f 690000 ffff0000 ffff ffff0000 1 1 0 0 0 130000 8021 0 0 0 10000 1 0 0 801f29 0 0 0 ff000000 1ff 0 0 0 0 13000000 0 da00d300 d400 0 1801f00 8f000000 ff000000 ffffff ff000000 1ff 100 0 0 0 13000000 802100 0 0 0 1000000 100 0 0 801f2900 0 0 0 0 1ffff 0 0 0 0 0 30013 a70000 a800dd 0 0 801f0000 1 8f ffffffff 0 1ffff 10000 0 0 0 0 590013 590000 590000 590000 0 67f0000 1e50000 ffff0000 ffffffff ffffffff 1 1 0 0 0 1 0 3 454d4305 25453 de028000 20 0'33 10001 0 0 0 0 3601 0 10000 0 0 0 0 0d@@iy(x)=( ( ~"`msub_((v),(Oy))")/( ~"`msub_((v),(Ox))") )*x - (1)/(2) *g*( (x)/( ~"`msub_((v),(Ox))") )^(2)Times New Roman dy(x)= ( ~"`msub_((v),(Oy))"*x)/( ~"`msub_((v),(Ox))") - (g*(x)^(2))/(2*( ~"`msub_((v),(Ox))")^(2))Times New Roman  *ArialArialArialArialArialSymbolArial+ proiettileplus proiettilecabrijava Le nostre risposte Attivit 4 Confrontate le risposte che ora avete fornito con quella di altri gruppi di lavoro. Le nostre risposte dopo il confronto con altri gruppi Problema 1 (da svolgersi a coppie) Un proiettile viene lanciato con una velocit di 20m/s con un angolo di 30 rispetto all'orizzontale da un'altezza di 30 metri sul livello del mare da una torre posta su un terreno pianeggiante. Quanto tempo rimane in aria, supponendo che non vi siano ostacoli? Qual la sua gittata? Qual la sua vlocit dopo 1 secondo? e dopo 3 secondi? Qual la massima altezza raggiunta dal proiettile? La nostra risposta Problema 2 (da svolgersi a coppie) Un proiettile viene lanciato su un terreno pianeggiante con una velocit di 30m/s con un angolo di 30 rispetto all'orizzontale da un'altezza di 0 metri sul livello del mare. Qual la minima distanza a cui si deve mettere un ostacolo alto 7 metri perch esso venga superato? Come cambia la risposta se l'angolo di tiro di 60? E se di 45? La nostra risposta Il moto di un proiettile ci ha fornito l'occasione per parlare della parabola come insieme di punti, ossia per descrivere la parabola con i metodi della geometria analitica. La parabola definita come l'insieme di tutti e soli i punti del piano che sono equidistanti da un punto (detto "fuoco" della parabola) e da una retta (detta "direttrice" della parabola. Il seguente file in cabriIIplus e il suo corrispondente in cabrijava consentono di fare qualche esplorazione su come cambia la parabola al variare del fuoco F e della direttrice d (nel caso del file in cabriIIplus potete anche vedere gli effetti che i cambiamenti hanno sull'equazione della parabola). Le modalit di costruzione dei file possono essere lette all'interno del file stesso. Attivit 5. (in piccoli gruppi) Esplorate uno dei due seguenti file (se avete Cabri II plus meglio cliccare sulla prima hotword, in quanto Cabri II plus consente un'esplorazione pi ricca) parabolafuocodirettriceplus parabolafuocodirettricejava I risultati della nostra esplorazione Attivit 6: l'equazione di una parabola (a coppie) Avete visto che si dice parabola il luogo geometrico dei punti P del piano che sono equidistanti da un punto F detto fuoco e da una retta d detta direttrice. Scegliete un sistema di riferimento che abbia l'asse delle y perpendicolare alla direttrice e che passi per il fuoco. L'asse x sia invece asse del segmento individuato, sull'asse y dal fuoco e dal punto di intersezione dell'asse y con la direttrice. Con una tale scelta del sistema di riferimento, il punto F ha coodinate (0; p), mentre la direttrice ha equazione y = -p. Considerate ora un generico punto P(x;y) del piano: esso, per quanto detto, appartiene alla parabola se e solo se la distanza di P dalla direttrice, ossia |y + p| uguale alla distanza PF. Dimostrate che, nel sistema di riferimento scelto, l'equazione della parabola di fuoco F e direttrice y = -p  La nostra dimostrazione E se il fuoco non fosse sull'asse delle y e avesse coordinate (q; p)? Se la direttrice una retta parallela all'asse delle x, di equazione y = d, la parabola ha equazione  Attivit 7. (individuale) Dimostra quanto detto in precedenza e cio che se il fuoco ha coordinate (q; p) e la direttrice una retta parallela all'asse delle x, di equazione y = d, la parabola ha equazione  La mia risposta Come potete notare, quando la direttrice una retta parallela all'asse delle ascisse, l'equazione della parabola assume la forma propria delle funzioni quadratiche, ossia la legge che consente di determinare, data una certa x, la corrispondente y, del tipo: f(x) = a*x^2 + b*x + c Attivit 8. (individuale) Ti chiediamo di seguire con attenzione il metodo che abbiamo utilizzato per determinare la relazione che lega i coefficenti a, b e c della funzione quadratica f(x) = a*x^2 + b*x + c alle coordinate del fuoco F(q;p) e all'equazione y= d della direttrice della parabola corrispondente.     Possiamo quindi dire che il fuoco di una parabola, grafico della funzione quadratica f(x) = a*x^2 + b*x + c, ha coordinate (-b/(2*a); (1- b^2+4*a*c)/(4*a)) , mentre la direttrice ha equazione y = (4*a*c - b^2 - 1)/(4*a) Determina ora tu le coordinate del vertice della parabola grafico di f(x) = a*x^2 + b*x + c in funzione di a,b,c La mia risposta E se la direttrice della parabola fosse una retta verticale, ossia parallela all'asse delle y? In questo caso si pu dimostrare che la sa equazine sarebbe del tipo x = a*y^2 + b*y + c Nota che tale equazione si ottiene da quella y = a*x^2+b*x+c con una simmetria rispetto alla bisettrice del primo e terzo quadrante che scambia x con y e y con x. Ci suggerisce che per determinare le coordinate del foco, del vertice e l'equazione della direttrice di parabole di questo tipo sia sufficiente scambiare fra di loro la x e la y nelle formule riferite alle parabole del tipo y=a*x^2+b*x+c. Cos come esiste il compasso per tracciare circonferenze ed esiste il meccanismo di Paucellier per tracciare rette, possibile costruire macchine matematiche che traccino parabole. Uno di questi parabolografi simulato nei seguenti file di cabri II plus e nel corrispondente file di cabrijava. parabolografocabriplus parabolografocabrijava Attivit 9 (in piccoli gruppi). Aprite uno dei due file (quello di Cabri II plus, se disponete di tale software) ed esplorate il funzionamento del meccanismo. In particolare, se aprite il file di Cabri II plus, guardate come varia l'equazione della parabola al variare della posizione della direttrice e del fuoco. I risultati delle nostre esplorazioni Esplorand il funzionamento del meccanismo articolato che genera parabole, vi sarete accorti che la diagonale BD del parabolografo tangente alla parabola. I problemi che vogliamo ora proporvi sono i seguenti: problema 3. Date le equazioni di una retta e di una parabola, come stabilire se esse sono o non sono tangenti? problema 4. Data l'equazione di una parabola e un suo punto P, come trovare l'equazione della retta tangente alla parabola in P? problema 5. Data l'equazione di una retta , come imporre che essa sia tangente a una parabola di cui si vuole determinare l'equazione? Tutti e tre i problemi sono ovviamente riconducibili a quello di determinare la condizione di tangenza fra una retta e una parabola. noto che i punti di intersezione fra una retta e una parabola si determinano risolvendo il sistema formato dall'equazione della retta, che di primo grado, e da quella della parabola, che di secondo grado. Si tratta quindi di un sistema di secondo grado, che pu avere al pi due soluzioni. Supponiamo che le equazioni della retta e della parabola siano del tipo: y = p*x + q y = a*x^2 + b*x + c L'equazione risolvente il sistema, ossia p*x + q = a*x^2 + b*x + c un'equazione di secondo grado in x (essa fornisce le ascisse dei punti di intersezione della retta con la parabola) del tipo A*x^2 + B*x + C=0, che pu avere due soluzioni, una soluzione, nessuna soluzione. Ricorderete che ci accade a seconda che la quantit B^2 - 4*A*C sia positiva, nulla o negativa. quindi abbastanza naturale comprendee che: a) se B^2 - 4*A*C > 0, la retta secante alla parabola b) se B^2 - 4*A*C = 0, la retta tangente alla parabola c) se B^2 - 4*A*C < 0, la retta esterna alla parabola La condizione di tangenza fra una retta e una parabola quindi espressa dall'equazione B^2 - 4*A*C = 0 , ossia dal porre uguale a 0 il cosiddetto "delta" dell'equazione risolvente il sistema fra l'equazione della retta e quella della parabola. Nel caso del problema 4., se l'equazione dlla parabola del tipo f(x) = a*x^2+b*x+c l'equazione della retta tangente pu essere determinata velocemente con la formula: y = f(x0) + f'(x0) (x- x0) Spiega perch (rispondi individualmente e poi confronta la tua risposta con quella di altri compagni) la mia risposta la mia risposta dopo il confronto con i miei compagni Scheda 3. Problemi sulla parabola (Si prevedono circa 6 ore di lavoro in classe) Vi chiediamo ora di leggere attentemente i testi dei seguenti problemi e di provare a risolverli in coppie. Confrontate poi la risoluzione che avete proposto con quella di almeno un'altra coppia di studenti. Chiedete al vostro insegnante eventuali chiarimenti su problemi eventualmente incontrati. Nei calcoli pi complessi aiutatevi pure con il manipolatore simbolico di TI-InterActive!, ma non esagerate: in alcuni casi si fa addirittura prima a fare i calcoli a mente e, in ogni caso, un po' di esercizio di calcolo "a mano" non pu che essere salutare! 1. Qual l'equazione della parabola P che interseca l'asse x nei punti di ascisse 2 e 3 e l'asse y nel punto di ascissa 6? Qual l'equazione della parabola che il fuoco nel vertice di P e il vertice nel fuoco di P? la nostra risposta la nostra risposta dopo il confronto con un'altra coppia 2. Determinate l'equazione cartesiana del luogo geometrico dei punti del piano di coordinate (2t - 1, t2 - t) la nostra risposta la nostra risposta dopo il confronto con un'altra coppia 3. Nella famiglia di parabole y = kx2 - 1, individuate quella tangente alla retta di equazione y = 2x - 2 la nostra risposta la nostra risposta dopo il confronto con un'altra coppia 4. Inscrivete un quadrato nel segmento parabolico individuato dall'asse y e dalla parabola di equazione x = y2 - 8 la nostra risposta la nostra risposta dopo il confronto con un'altra coppia 5. Un punto materiale P viene lanciato da unaltezza di 100m. in un campo gravitazionale di intensit g con una velocit iniziale di intensit 500 m/s e formante con il vettore accelerazione gravitazionale un angolo di 135. Dopo aver descritto le leggi orarie di P (che soggetto alla sola forza di gravit), determinate: a) la traiettoria di P b) la gittata c) il tempo che rimane in aria d) il tempo in cui raggiunge la massima altezza e) la massima altezza la nostra risposta la nostra risposta dopo il confronto con un'altra coppia 6. Determinate le equazioni delle parabole aventi asse di simmetria parallelo all'asse y, passanti per A(1,0) , B(8,0) e tangenti alla retta y + 5x - 4 = 0 la nostra risposta la nostra risposta dopo il confronto con un'altra coppia Prima di eseguire il solito test si autovalutazione, gli esercizi di consolidamento e le prove di verifica, ti chiediamo di fare tu, insieme a qualche compagno di gruppo, il documento di sintesi teorica di questa lezione. Confrontalo poi con quello proposto da altri compagni di gruppo e, infine, proponete la vostra sintesi all'insegnante. La nostra sintesi Test di autovalutazione Consolidamento Esempi di verifiche xB @P@P)*@P,P?@PO PZTP@P6P@P PP@PP@P@P PP[@P6PJ%@Po@Pq@PA PNP@P@P"=>@P@'Pg@Px(P P@P5 @P6 N@P @P A@P @P 5@P @P 1@P. @P/ @P <@P +@P <@P @P P @P P P @P P @P@P@P@P'@P PP@P<@P @P@PA@P@P@PcPs~@P@P !"@P% P/PE@Pa&P@Pc PneP@P PxPV P`}Pe@PB@PC^@P@P8@P@P<@P@P@P@PK@P@P@P@P@P@P@P@Pj@PnP~@P6P @P!P@P@Pj"@Pl"@P"@P"%@P"@P"W@P"@P#@P#@P#@P#P1# @P<#9Pu#g@P#@P#@P#@P#@P#2@P#@P#@P#@P#@P#@P#P$@P$9PM$)@Pv$@Pw$@Pz$@P|$2@P}$@P~$@P$8@P$@P$@P$@P$@P$@P$@P$P$ @P$9P%R@Pq%@Pr%@P%@P%@P%@P%2@P%@P%@P%@P%P%@P%9P%'@P &@P!&P@Pq&@Pr&@P'@P'Q@Pe'@Pg's@P'P'@P(9P<(d@P(@P(@P(@P(@P(@P(@P(@P(@P(@P(@P(P )@P)9PU)e@P* P*P* * *P*++P ++"Arial"Arial"Arial"ArialSymbol"Arial*++,>??@@AABBCCDDEEFFGGHHIIJJKKLLMMNNOYZ5667IJJKKLLMMNNOOPPQ@AAB  !>??@@AfgghhiijjkkllmmnnooppqqrrssttuuvvwwxH I B C  z {                                      34KL`aabbcrssttuuvvwwx12   "##$$%DE_``a,-bccdTUUV78  IJ?@@Almmn}~~  C"D"D"E"##0#1#1#2#2#3#3#4#4#5#5#6#6#7#7#8#8#9#9#:#:#;#;#<#t#u#v#w#w#x#x#y#y#z#z#{#{#|#|#}###$$$$$$$$$$$$$ $ $ $ $ $ $ $ $ $ $$$$$$$$$$$$$$L$M$N$O$O$P$P$Q$Q$R$R$S$S$T$T$U$V$W$W$X$$$$$$$$$$$$$$$$$$$$$$$$$$$%% %!%!%"%"%#%#%$%$%%%%%&%&%'%(%)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&&&&&&&&&&&&&&&& & & &&&N'O'f'g't'u''''''''''''''''''''''''''''''''''''''''''''''(((((((;(<(=(>(>(?(?(@(@(A(A(B(B(C(C(D(D(E(E(F(F(G(H(I((((() ) ) ) ) ) ) ) ) ) ))))))))))))))))))))))))))))))T)U)V)W)W)X)Y)Z)Z)[)[)\)\)])])^)^)_)_)`)`)a)a)b)b)c)**************************************+++++++d p@ P !$`'dL  \ l0!#dL  \ l0!#dL  \ l0!#&dL  \ l0!#&dL  \ l0!#@@ddH;.7. 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secondo? e dopo 3 secondi? Qual la massima altezza raggiunta dal proiettile? La nostra risposta Problema 2 (da svolgersi a coppie) Un proiettile viene lanciato su un terreno pianeggiante con una velocit di 30m/s con un angolo di 30 rispetto all'orizzontale da un'altezza di 0 metri sul livello del mare. Qual NANIa. Il seguente file in cabriIIplus e il suo corrispondente in cabrijava consentono di fare qualche esplorazione su come cambia la parabola al variare del fuoco F e della direttrice d (nel caso del file in cabriIIplus potete anche vedere gli effetti che i cambiamenti hanno sull'equazione della parabola). Le modalit di costruzione dei file possono essere lette all'interno del file stesso. Attivit 5. (in piccoli gruppi) Esplorate uno dei due seguenti file (se avete Cabri II plus lkN@dd ;<7< CTxobjItem<ࡱ>  Root EntryZn@3jd@ContentsG)OlePres000g 1456 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 88656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 0 ff00ff00 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff0000 ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 0 ffff00ff 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 ffffff88 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff0000 ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 0 ffff00ff 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff0000 ffffff 78 64 64745305 14954 10000 0 8001 0 0 0 2bc0000 0 70100 49540000 74614d20 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