Fin des corrections de quelques équations de balistique
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This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex 2021.1.1) 2 MAY 2021 21:46
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This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex 2021.1.1) 5 MAY 2021 08:43
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@ -1960170,14 +1960170,14 @@ FD(\014)t FJ(\))24 b(=)f(0)116 4805 y FW(o\371)28 b(on)f(a)g
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FJ(\))p FD(=)14 b FJ(cos\()p FD(\014)t FJ(\))116 5144
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y FW(En)26 b(mettan)n(t)h(en)f(\351vidence)g FD(T)1051
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5156 y FG(1)1114 5144 y FW(et)h(le)f(d\351pla\347an)n(t)g(\340)g
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(droite)116 5244 y(de)i(l'\351quation,)f(on)g(a)h(:)487
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5414 y FD(F)j FH(\000)18 b FD(P)35 b FJ(=)22 b FD(T)878
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5426 y FG(1)933 5414 y FH(\001)d FJ(\(sin)q(\()p FD(\013)p
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5414 y FD(F)i FH(\000)18 b FD(P)35 b FJ(=)23 b FD(T)725
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5426 y FG(1)780 5414 y FH(\001)18 b FJ(\(sin)q(\()p FD(\013)p
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FJ(\))i(+)e(cos)o(\()p FD(\013)p FJ(\))h FH(\001)g FJ(tan\()p
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@ -1960456,28 +1960456,29 @@ showpage
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@endspecial 0 TeXcolorgray 2034 2307 a(et)g(donc)f(\034nalemen)n(t)h
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(a)n(v)n(ec)e FD(P)35 b FJ(=)23 b FD(m)18 b FH(\001)g
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FD(g)31 b FW(:)2573 2549 y FD(T)2622 2561 y FG(1)2682
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FD(g)31 b FW(:)2419 2549 y FD(T)2468 2561 y FG(1)2528
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2549 y FJ(=)2879 2492 y FD(F)f FH(\000)18 b FD(m)g FH(\001)h
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FD(g)p 2779 2530 541 4 v 2779 2606 a FJ(sin)q(\()p FD(\013)p
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FJ(\))g(+)f(sin)q(\()p FD(\014)t FJ(\))2034 2795 y FW(et)28
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b(p)r(our)f FD(T)2376 2807 y FG(2)2441 2795 y FW(:)2397
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3042 y FD(T)2446 3054 y FG(2)2506 3042 y FJ(=)2702 2986
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y(\()p FD(F)k FH(\000)18 b FD(m)g FH(\001)g FD(g)s FJ(\))h
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FH(\001)f FJ(cos)o(\()p FD(\013)p FJ(\))p 2604 3023 893
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4 v 2604 3099 a(cos)o(\()p FD(\014)t FJ(\))h FH(\001)g
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FJ(\(sin\()p FD(\013)p FJ(\))h(+)e(sin\()p FD(\014)t
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FJ(\)\))2034 3368 y FK(4.4.2)113 b(Plan)37 b(inclin\351)2117
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3541 y FW(A)26 b(l'instar)f(de)h(la)g(c)n(h)n(ute)f(libre,)h(on)f(p)r
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(eut)i(consid\351rer)d(le)2034 3640 y(probl\350me)j(historique)2766
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3610 y FU(a)2831 3640 y FW(suiv)-5 b(an)n(t)27 b(:)2117
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3750 y(d\351terminez)56 b(l'acc\351l\351ration)f(d'une)i(masse)e
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FD(m)i FW(qui)2034 3850 y(glisse)33 b(le)h(long)f(d'un)h(plan)f
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(inclin\351)h(faisan)n(t)f(un)i(angle)d FD(\013)2034
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3950 y FW(a)n(v)n(ec)26 b(l'horizon)n(tale.)g(On)i(n\351glige)e(les)i
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(frottemen)n(ts.)2034 4049 y(R\351p)r(onse)f(:)2117 4159
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y(Comme)34 b(le)g(mon)n(tre)g(la)g(\034gure)f(4.2,)h(deux)g(forces)f
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(ex-)2034 4259 y(t\351rieures)38 b(seulemen)n(t)h(s'exercen)n(t)f(sur)g
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(la)h(masse)f FD(m)p FW(.)h(Il)2034 4359 y(s'agit)27
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FD(g)p 2626 2530 848 4 v 2626 2606 a FJ(sin\()p FD(\013)p
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FJ(\))h(+)e(cos)o(\()p FD(\013)p FJ(\))h FH(\001)g FJ(tan)q(\()p
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FD(\014)t FJ(\))2034 2795 y FW(et)28 b(p)r(our)f FD(T)2376
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2807 y FG(2)2441 2795 y FW(:)2244 3042 y FD(T)2293 3054
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y FG(2)2352 3042 y FJ(=)2702 2986 y(\()p FD(F)k FH(\000)18
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b FD(m)g FH(\001)g FD(g)s FJ(\))h FH(\001)f FJ(cos)o(\()p
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FD(\013)p FJ(\))p 2450 3023 1200 4 v 2450 3099 a(cos)o(\()p
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FD(\014)t FJ(\))i FH(\001)e FJ(\(sin)q(\()p FD(\013)p
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FJ(\))h(+)f(cos\()p FD(\013)p FJ(\))h FH(\001)g FJ(tan\()p
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FD(\014)t FJ(\)\))2034 3368 y FK(4.4.2)113 b(Plan)37
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b(inclin\351)2117 3541 y FW(A)26 b(l'instar)f(de)h(la)g(c)n(h)n(ute)f
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(libre,)h(on)f(p)r(eut)i(consid\351rer)d(le)2034 3640
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y(probl\350me)j(historique)2766 3610 y FU(a)2831 3640
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y FW(suiv)-5 b(an)n(t)27 b(:)2117 3750 y(d\351terminez)56
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b(l'acc\351l\351ration)f(d'une)i(masse)e FD(m)i FW(qui)2034
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3850 y(glisse)33 b(le)h(long)f(d'un)h(plan)f(inclin\351)h(faisan)n(t)f
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(un)i(angle)d FD(\013)2034 3950 y FW(a)n(v)n(ec)26 b(l'horizon)n(tale.)
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g(On)i(n\351glige)e(les)i(frottemen)n(ts.)2034 4049 y(R\351p)r(onse)f
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(:)2117 4159 y(Comme)34 b(le)g(mon)n(tre)g(la)g(\034gure)f(4.2,)h(deux)
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g(forces)f(ex-)2034 4259 y(t\351rieures)38 b(seulemen)n(t)h(s'exercen)n
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(t)f(sur)g(la)h(masse)f FD(m)p FW(.)h(Il)2034 4359 y(s'agit)27
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b(de)h(son)g(p)r(oids)2736 4302 y FH(\000)-65 b(!)2745
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2034 4470 y(clin\351)2226 4413 y FH(\000)-65 b(!)2236
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@ -1961344,11 +1961345,11 @@ FJ(sin\(2)18 b FH(\001)h FD(\013)p FJ(\))24 b(=)1454
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3959 a(v)1536 3935 y FG(2)1533 3979 y FC(o)266 4125 y
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FH(\))46 b FD(\013)24 b FJ(=)f(arcsin)o(\()815 4069 y
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(l'angle)f(n\351cessaire)f(p)r(our)h(at-)91 4433 y(teindre)34
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b(le)h(p)r(oin)n(t)f(C)g(de)g(co)r(ordonn\351e)f FJ(\()p
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FD(x)1430 4445 y FG(1)1468 4433 y FJ(;)14 b FD(y)1546
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b(d\351terminer)i(l'angle)f(n\351cessaire)f(p)r(our)h(at-)91
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4433 y(teindre)34 b(le)h(p)r(oin)n(t)f(C)g(de)g(co)r(ordonn\351e)f
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FJ(\()p FD(x)1430 4445 y FG(1)1468 4433 y FJ(;)14 b FD(y)1546
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4445 y FG(1)1583 4433 y FJ(\))p FW(,)34 b(il)91 4533
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y(faut)24 b(partir)f(de)h(l'\351quation)f(de)h(la)g(parab)r(ole)e
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(appli-)91 4632 y(qu\351e)28 b(au)f(p)r(oin)n(t)h(C)f(:)119
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@ -195,11 +195,11 @@ et on remplace dans la seconde~:
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où on a utilisé la définition de la tangente~:
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\[\tan(\beta)=\sin(\beta)/\cos(\beta)\]
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En mettant en évidence \(T_1\) et le déplaçant à droite de l'équation, on a~:
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\[F-P=T_1\cdot (\sin(\alpha)+\sin(\beta))\]
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\[F-P=T_1\cdot (\sin(\alpha)+\cos(\alpha)\cdot\tan(\beta))\]
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et donc finalement avec \(P=m\cdot g\)~:
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\[T_1=\frac{F-m\cdot g}{\sin(\alpha)+\sin(\beta)}\]
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\[T_1=\frac{F-m\cdot g}{\sin(\alpha)+\cos(\alpha)\cdot\tan(\beta)}\]
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et pour \(T_2\)~:
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\[T_2=\frac{(F-m\cdot g)\cdot \cos(\alpha)}{\cos(\beta)\cdot (\sin(\alpha)+\sin(\beta))}\]
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\[T_2=\frac{(F-m\cdot g)\cdot \cos(\alpha)}{\cos(\beta)\cdot (\sin(\alpha)+\cos(\alpha)\cdot\tan(\beta))}\]
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\subsection{Plan incliné}
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\begin{figure}[t]
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@ -359,7 +359,7 @@ Car, \(2\cdot \sin(\alpha)\cos(\alpha)=\sin(2\cdot \alpha)\).\\
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Une conséquence de ce résultat est qu'on peut déterminer l'angle sous lequel on doit tirer pour que le tir une porté L~:
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\begin{align*}
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L&=\frac{v_o^2}{g}\cdot \sin(2\cdot \alpha)\;\Rightarrow\;\sin(2\cdot \alpha)=\frac{g\cdot L}{v_o^2}\\
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&\Rightarrow\;\alpha = \arcsin(\frac{g\cdot L}{v_o^2})
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&\Rightarrow\;\alpha = \arcsin(\frac{g\cdot L}{v_o^2})/2
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\end{align*}
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\item Pour déterminer l'angle nécessaire pour atteindre le point C de coordonnée \((x_1;y_1)\), il faut partir de l'équation de la parabole appliquée au point C~:
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\[y_1=-\frac{1}{2}\cdot g\cdot\left(\frac{x_1}{v_o}\right)^2\cdot \frac{1}{\cos^2(\alpha)}+x_1\cdot \tan(\alpha)\]
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