1_3_MRUV_Parte_1.ipynb

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    "**Autor: Gustavo Henrique Reis de Araújo Lima**\n",
    "\n",
    "**Instituição: CEFET - MG - Unidade Contagem**\n",
    "\n",
    "**(feito em Julia 1.4)**\n",
    "\n",
    "**Última atualização: 05/05/2020**\n",
    "\n",
    "**Requisitos: Plots.jl, LaTeXStrings.jl**\n",
    "\n",
    "***\n",
    "# **1. Cinemática**\n",
    "\n",
    "## **1.3  Movimento Retilíneo Uniformemente Variado (MRUV)**\n",
    "\n",
    "### **1.3.1  Velocidade em função do tempo no MRUV**\n",
    "\n",
    "O tipo de movimento mais simples possível é o movimento onde a velocidade é constante: movimento retilíneo uniforme. O próximo passo é entender o que acontece com o movimento quando a velocidade não é mais constante.\n",
    "\n",
    "Por enquanto, o movimento continua sendo retilíneo: a direção do movimento não muda. Mas vamos considerar que o módulo e o sentido da velocidade podem variar de forma uniforme. Esse tipo de movimento é chamado de movimento retilíneo uniformemente variado (MRUV).\n",
    "\n",
    "Quando dizemos que a velocidade varia de maneira uniforme, isso significa que a velocidade sempre varia a mesma quantidade $\\Delta v$ em qualquer intervalo de tempo igual a $\\Delta t$. O mesmo tipo de comportamento que ocorria com a posição no MRU.\n",
    "\n",
    "A figura abaixo mostra um exemplo de velocidade que varia uniformemente com o tempo no MRUV."
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   "source": [
    "using Plots\n",
    "using LaTeXStrings\n",
    "using Printf\n",
    "\n",
    "#definindo a função v(t): a é a aceleração, v0 é a velocidade inicial, t é o instante de tempo\n",
    "function vel(a::Real, v0::Real, t::Real)\n",
    "    return a*t + v0\n",
    "end\n",
    "\n",
    "#definindo características padrões para os gráficos\n",
    "default(size = [800,600], grid = true, markersize = 8, framestyle = :origin, linewidth = 3, legend = :false)\n",
    "\n",
    "#definindo vetor com os valores de t utilizados para desenhar o gráfico\n",
    "t = 0:1:20\n",
    "tmax = maximum(t)\n",
    "\n",
    "#definindo vetor com as velocidades utilizadas para desenhar o gráfico\n",
    "a = 3\n",
    "v0 = 10\n",
    "vx = vel.(a,v0,t)\n",
    "vxmax = maximum(vx)\n",
    "\n",
    "#desenhando o gráfico\n",
    "plot(t,vx, xaxis = (\"tempo (s)\", (0, tmax), 0:2:tmax, font(10)),\n",
    "   yaxis = (\"velocidade (m/s)\", (0,vxmax), v0:2*a:vxmax, font(10)), title = \"velocidade x tempo\")"
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   "source": [
    "Novamente, temos uma reta representando uma grandeza em função do tempo. Toda vez que uma grandeza varia uniformemente em função de outra, o gráfico que representa essa variação é uma reta.\n",
    "\n",
    "Vamos aplicar o mesmo tipo de tratamento matemático que aplicamos ao gráfico de posição x tempo no MRU, mas no caso do MRUV, aplicando ao gráfico de velocidade x tempo. Lembrando que a equação de um gráfico linear como o mostrado acima é do tipo: $f(x) = a\\cdot x + b$. Como temos a velocidade $v$ em função do tempo $t$, a equação será escrita como:\n",
    "\n",
    "$$ v(t) = a\\cdot t + b.$$\n",
    "\n",
    "O coeficiente $a$ é o coeficiente angular ou inclinação da reta que representa a função $v(t)$. O coeficiente $b$ é igual à velocidade do objeto no instante $t=0$, ou seja, $b$ é a velocidade inicial $v_0$ do objeto. \n",
    "\n",
    "Para entender o significado da inclinação do gráfico nesse caso, vamos comparar algumas velocidades de objetos em MRUV."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#desenhando gráficos de velocidade x tempo no MRUV com várias inclinações diferentes.\n",
    "\n",
    "plot(t,vx, xaxis = (\"tempo (s)\", (0, tmax), 0:2:tmax, font(10)),\n",
    "    yaxis = (\"velocidade (m/s)\", (-vxmax,vxmax), -vxmax:2*a:vxmax, font(10)), title = \"velocidade x tempo\",\n",
    "    label = latexstring(\"{\\\\rm Objeto ~ 1:} ~ v_1(t)\"), legend = :bottomleft, legendfontsize = 10)\n",
    "plot!(t,vel.(2*a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 2:} ~v_2(t)\"))\n",
    "plot!(t,vel.(a/2,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 3:} ~v_3(t)\"))\n",
    "plot!([0,tmax],[v0,v0], label = latexstring(\"{\\\\rm Objeto ~ 4:} ~v_4(t)\"))\n",
    "plot!(t,vel.(-a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 5:} ~v_5(t)\"))\n",
    "plot!(t,vel.(-1.5*a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 6:} ~v_6(t)\"))\n",
    "plot!([8,8],[-vxmax,vxmax], lw = 1, ls = :dash, color = :black, label = :none)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Todos os objetos têm a mesma velocidade inicial $v_0$ para simplificar a análise da inclinação. A velocidade do objeto 2 é a que crescer mais rápida entre todos os objetos representados. Como a velocidade cresce a medida que o tempo passa, dizemos que a reta que a representa tem inclinação positiva, portanto, a reta que representa a velocidade do objeto 2 tem a maior inclinação. Já a velocidade do objeto 6 é a que decresce mais rapidamente entre todos os objetos. Quando a velocidade diminui a medida que o tempo passa, dizemos que a reta que a representa tem inclinação negativa, portanto, a reta que representa a velocidade do objeto 6, tem a menor inclinação. A velocidade do objeto 4 é constante, portanto, a reta que a representa tem inclinação nula, ou seja, o objeto está em MRU.\n",
    "\n",
    "A grandeza que representa o quanto a velocidade de um objeto varia com o tempo é chamada de *aceleração*, e também é uma grandeza vetorial representada como $\\vec{a}$. Lembrando que, enquanto estivermos analisando movimentos retilíneos, onde não há mudança de direção no movimento, não precisamos nos preocupar com a notação e nem a realidade vetorial das grandezas, por isso vamos representar a aceleração apenas como $a$.\n",
    "\n",
    "Assim sendo, a inclinação do gráfico é igual a aceleração $a$ do movimento, e a equação que representa a velocidade em função do tempo pode ser escrita como:\n",
    "\n",
    "<div class=\"alert alert-block alert-warning\">\n",
    "    <b>Equação da velocidade em função do tempo (MRUV):</b> \n",
    "    $$v(t) = a\\cdot t + v_0.$$\n",
    "</div>\n",
    "\n",
    "A utilização da equação acima nos permite descobrir a velocidade do objeto em qualquer instante do movimento. A velocidade do objeto em um instante $t$ qualquer é chamada de *velocidade instantânea*."
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     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#desenhando gráficos de velocidade x tempo no MRUV com várias inclinações diferentes.\n",
    "\n",
    "plot(t,vx, xaxis = (\"tempo (s)\", (0, tmax), 0:2:tmax, font(10)),\n",
    "    yaxis = (\"velocidade (m/s)\", (-vxmax,vxmax), -vxmax:2*a:vxmax, font(10)), title = \"velocidade x tempo\",\n",
    "    label = latexstring(\"{\\\\rm Objeto ~ 1:} ~ a_1 > 0\"), legend = :bottomleft, legendfontsize = 10)\n",
    "plot!(t,vel.(2*a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 2:} ~ a_2 > a_1 > 0\"))\n",
    "plot!(t,vel.(a/2,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 3:} ~ a_1 > a_3 > 0\"))\n",
    "plot!([0,tmax],[v0,v0], label = latexstring(\"{\\\\rm Objeto ~ 4:} ~ a_4 = 0\"))\n",
    "plot!(t,vel.(-a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 5:} ~ a_5 < 0\"))\n",
    "plot!(t,vel.(-1.5*a,v0,t), label = latexstring(\"{\\\\rm Objeto ~ 6:} ~ a_6 < a_5 < 0\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3.2 Aceleração ($\\vec{a}$) no MRUV\n",
    "\n",
    "**No MRUV, a velocidade do objeto varia, mas com aceleração constante!**\n",
    "\n",
    "Para se calcular a aceleração do objeto (*inclinação da reta*) a partir do gráfico de velocidade em função do tempo temos que:\n",
    "\n",
    "* escolher dois instantes de tempo $t_1$ e $t_2$ quaisquer, nos quais sejam conhecidas as velocidades do objeto $v_1$ e $v_2$ nos instantes respectivos;\n",
    "* calcular a variação da velocidade $\\Delta v = v_2 - v_1$, e o intervalo de tempo no qual ocorreu essa variação de velocidade $\\Delta t = t_2 - t_1$;\n",
    "* por fim, a inclinação é dada por:\n",
    "\n",
    "$$ a = \\frac{\\Delta v}{\\Delta t}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#escolhendo instantes de tempo t1 e t2 que serão utilizado no cálculo da aceleração\n",
    "t1 = 6; t2 = 17;\n",
    "Δt = t2 - t1\n",
    "\n",
    "#calculando as velocidades nos respectivos instantes de tempo\n",
    "v11 = vel(a,v0,t1)\n",
    "v12 = vel(a,v0,t2)\n",
    "Δv = v12 - v11\n",
    "\n",
    "#calculando a aceleração do objeto\n",
    "a1 = Δv/Δt\n",
    "\n",
    "#criando strings para anotações\n",
    "#string para Δt\n",
    "dtstr = @sprintf(\"%3.1f\", Δt)\n",
    "dttext = latexstring(\"\\\\Delta t = \", dtstr, \"\\\\,{\\\\rm s}\")\n",
    "dttext = replace(dttext, \".\" => \",\\\\!\")\n",
    "#definindo posição da anotação sobre Δt\n",
    "dttext_pos = [(t2+t1)/2, 0.9*v11]\n",
    "\n",
    "#strings para Δv\n",
    "dvstr = @sprintf(\"%3.1f\", Δv)\n",
    "dvtext = latexstring(\"\\\\Delta v = \", dvstr, \"\\\\,{\\\\rm m/s}\")\n",
    "dvtext = replace(dvtext, \".\" => \",\\\\!\")\n",
    "#definindo posição da anotação sobre Δv\n",
    "dvtext_pos = [1.03*t2, (v11+v12)/2]\n",
    "\n",
    "#string para a\n",
    "a1str = @sprintf(\"%3.2f\", a1)\n",
    "a1text = latexstring(\"a = \\\\frac{\\\\Delta v}{\\\\Delta t} = \\\\frac{\", dvstr, \"\\\\,{\\\\rm m/s}}{\", \n",
    "    dtstr, \"\\\\,{\\\\rm s}} =\", a1str, \"\\\\,{\\\\rm m/s^2}\")\n",
    "a1text = replace(a1text, \".\" => \",\\\\!\")\n",
    "#definindo posição da anotação sobre a\n",
    "a1text_pos = [3, 0.95*vxmax]\n",
    "\n",
    "#criando vetor de anotações\n",
    "anots = [(dttext_pos[1], dttext_pos[2], text(dttext, halign = :center, pointsize = 12, color = :blue)),\n",
    "    (dvtext_pos[1], dvtext_pos[2], text(dvtext, valign = :center, pointsize = 12, color = :blue, rotation = 90)),\n",
    "    (a1text_pos[1], a1text_pos[2], text(a1text, halign = :left, pointsize = 12, color = :blue))]\n",
    "\n",
    "#desenhando gráfico de v x t\n",
    "plot(t,vx, xaxis = (\"tempo (s)\", (0, tmax), [0,t1,t2, tmax], font(10)),\n",
    "    yaxis = (\"velocidade (m/s)\", (0,vxmax), [0,v0,v11,v12,vxmax], font(10)), title = \"velocidade x tempo\")\n",
    "\n",
    "#desenhando pontos escolhidos no gráfico\n",
    "scatter!([t1,t2],[v11,v12], marker = :circle, color = :red, alpha = 0.5)\n",
    "\n",
    "#marcando ponto t1, v1 no gráfico\n",
    "plot!([0,t1],[v11,v11], color = :red, alpha = 0.3, ls = :dash)\n",
    "plot!([t1,t1],[0,v11], color = :red, alpha = 0.3, ls = :dash)\n",
    "\n",
    "#marcando ponto t2,v2 no gráfico\n",
    "plot!([0,t2],[v12,v12], color = :red, alpha = 0.3, ls = :dash)\n",
    "plot!([t2,t2],[0,v12], color = :red, alpha = 0.3, ls = :dash)\n",
    "\n",
    "#marcando comprimentos de Δt e Δv no gráfico\n",
    "plot!([t1,t2],[v11,v11], color = :dodgerblue, alpha = 0.5, ls = :dash)\n",
    "plot!([t2,t2],[v12,v11], color = :dodgerblue, alpha = 0.5, ls = :dash)\n",
    "\n",
    "#anotando valores no gráfico\n",
    "annotate!(anots)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que a aceleração $a$ é a mesma para quaisquer instantes $t_1$ e $t_2$ que forem escolhidos.\n",
    "\n",
    "A equação acima é utilizada, na verdade, no cálculo da aceleração média $a_{\\rm med}$ de um objeto durante um intervalo de tempo $\\Delta t$. Entretanto, quando um objeto está em MRUV, sua aceleração é constante e, assim sendo, será igual à sua aceleração média durante todo o movimento.\n",
    "\n",
    "Já no caso de movimentos não-retilíneos, onde a direção do objeto também varia com o tempo, deve-se levar em consideração a identidade vetorial das grandezas e, então, a aceleração média é:\n",
    "\n",
    "<div class=\"alert alert-block alert-warning\">\n",
    "    <b>Aceleração média:</b> \n",
    "    $$\\overrightarrow{a}_{\\rm \\! med} = \\frac{\\overrightarrow{\\Delta v}}{\\Delta t},$$\n",
    "</div>\n",
    "\n",
    "e, dificilmente, esta será igual a aceleração no instante $t_1$ ($\\overrightarrow{a}_{\\! 1}$) , ou a aceleração no instante $t_2$ ($\\overrightarrow{a}_{\\! 2}$).\n",
    "\n",
    "A próxima figura mostra a representação gráfica da aceleração em função do tempo no MRUV."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "ytitle = string(\"aceleração (m/s²)\")\n",
    "#ytitle = latexstring(\"\\\\textrm{acelera\\\\c{c}\\\\~{a}o} \\\\, ({\\\\rm m/s^2})\")\n",
    "\n",
    "plot([0,tmax],[a1,a1], xaxis = (\"tempo (s)\", (0, tmax), [0,20], font(10)), \n",
    "    yaxis = (ytitle, (0, 2*a1), [0,a1,2*a1], font(10)), color = :forestgreen,\n",
    "    title = \"aceleração x tempo\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A área entre a reta que representa a aceleração em função do tempo e o eixo do tempo, limitada por dois instantes de tempo $t_1$ e $t_2$ é igual à variação da velocidade no intervalo de tempo $\\Delta t = t_2 - t_1$. No caso do MRUV, a figura formada é um retângulo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#cálculo da área do retângulo\n",
    "dv = a1 * Δt\n",
    "\n",
    "#gerando string com valor da aceleração - altura do retângulo\n",
    "aceltext = latexstring(\"a = \", a1str, \"\\\\,{\\\\rm m/s^2}\")\n",
    "aceltext = replace(aceltext, \".\" => \",\\\\!\")\n",
    "\n",
    "#gerando string com o cálculo da área do retângulo\n",
    "areastr = @sprintf(\"%3.1f\", dv)\n",
    "areatext = latexstring(\"{\\\\rm \\\\acute{a}rea} = \", a1str, \"\\\\,{\\\\rm \\\\frac{m}{s^2}} \\\\cdot \", dtstr, \"\\\\,{\\\\rm s} = \",\n",
    "    areastr, \"\\\\,{\\\\rm \\\\frac{m}{s}}\")\n",
    "areatext = replace(areatext, \".\" => \",\\\\!\")\n",
    "\n",
    "#gerando anotações no gráfico\n",
    "anots = [(dttext_pos[1], 1.05*a1, text(dttext, halign = :center, pointsize=14, color = :green)),\n",
    "    (0.92*t1, a1/2, text(aceltext, valign = :center, pointsize = 14, color = :green, rotation = 90)),\n",
    "    (2, 1.3*a, text(areatext, halign = :left, pointsize = 14, color = :green))]\n",
    "\n",
    "#título do eixo y\n",
    "ytitle = string(\"aceleração (m/s²)\")\n",
    "#desenhando aceleração em função do tempo\n",
    "plot([0,tmax],[a1,a1], xaxis = (\"tempo (s)\", (0, tmax), [0,t1,t2,20], font(10)), \n",
    "    yaxis = (ytitle, (0, 2*a1), [0,a1,2*a1], font(10)), color = :forestgreen,\n",
    "    title = \"aceleração x tempo\", alpha = 0.5)\n",
    "\n",
    "#desenhando aceleração entre t1 e t2\n",
    "plot!([t1,t2],[a1,a1], color = :forestgreen, fill = (0, :palegreen), fillalpha = 0.5)\n",
    "\n",
    "#marcando instantes t1 e t2 no gráfico\n",
    "scatter!([t1,t2],[a1,a1], marker = :circle, color = :red, alpha = 0.3)\n",
    "plot!([t1,t1],[0,a1], ls = :dash, color = :forestgreen)\n",
    "plot!([t2,t2],[0,a1], ls = :dash, color = :forestgreen)\n",
    "\n",
    "#escrevendo anotações\n",
    "annotate!(anots)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A variação da velocidade encontrada pela área no gráfico de aceleração em função do tempo, é exatamente igual a variação da velocidade obtida pela análise do gráfico de velocidade em função do tempo, no mesmo intervalo observado.\n",
    "\n",
    "A equação do cálculo da aceleração média:\n",
    "\n",
    "$$ a_{\\rm med} = \\frac{\\Delta v}{\\Delta t},$$ \n",
    "\n",
    "pode ser reescrita da seguinte forma:\n",
    "\n",
    "$$ \\Delta v = a_{\\rm med} \\cdot \\Delta t,$$\n",
    "\n",
    "que é igual a área representada acima. Vale lembrar que **essa equação é válida apenas para MRUV**. Se o objeto não tiver aceleração constante a expressão para se calcular a variação da velocidade (*área*) será diferente.\n",
    "\n",
    "Apesar de ser possível descobrir a variação da velocidade $\\Delta v$ em um intervalo $\\Delta t$ através da análise do gráfico da aceleração em função do tempo, apenas com essas informações ainda não é possível descobrir a equação de $v(t)$. Também não é possível obter a informação necessária para se obtê-la apenas com a análise do gráfico de aceleração em função do tempo. Para isso, é necessário se conhecer a velocidade do objeto em um instante apenas, qualquer instante. Sem essa informação, existem infinitas equações possíveis com a mesma aceleração. \n",
    "\n",
    "Vimos algo semelhante quando falamos sobre a velocidade no MRU, com alguns exemplos de como se obter a equação de posição em função do tempo, dada a posição em um instante qualquer. O problema matemático aqui é exatamente igual, mudamos apenas as grandezas: velocidade e posição no MRU por aceleração e velocidade no MRUV.\n",
    "\n",
    "Como sabemos a velocidade inicial do objeto analisado, é a aceleração do objeto, também sabemos a equação de sua velocidade em função do tempo, e assim é possível calcular a velocidade do objeto em qualquer instante que quisermos.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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       "\" transform=\"translate(919, 766)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#criando string com v(t)\n",
    "v0str = @sprintf(\"%3.1f\", v0)\n",
    "veqtext = latexstring(\"v_1(t) = \", a1str, \"\\\\,{\\\\rm \\\\frac{m}{s^2}} \\\\cdot t + \", v0str, \"\\\\,{\\\\rm \\\\frac{m}{s}}\")\n",
    "veqtext = replace(veqtext, \".\" => \",\\\\!\")\n",
    "\n",
    "annoteq = [(5,(v0+5+5*a), text(veqtext, halign = :left, pointsize = 14, color = :blue, rotation = 30))]\n",
    "\n",
    "#desenhando gráfico de v x t\n",
    "plot(t,vx, xaxis = (\"tempo (s)\", (0, tmax), 0:5:20, font(10)),\n",
    "    yaxis = (\"velocidade (m/s)\", (0,vxmax), v0:5*a:vxmax, font(10)), title = \"velocidade x tempo\")\n",
    "annotate!(annoteq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.4.0",
   "language": "julia",
   "name": "julia-1.4"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.4.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}