Брзина код РППК и график брзине

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\begin{document}

\begin{center}
  {\large\bfseries Кинематика} \\[2pt]
  {\small Брзина код РППК и график брзине}
\end{center}

\section*{Тренутна брзина}

Из $a = \tfrac{v - v_0}{t}$ следи $v - v_0 = a \cdot t$, тј.

\[
  v = v_0 + a \cdot t
\]

Ако се $a$ пише са знаком, формула је увек иста: $a > 0$ за убрзано, $a < 0$ за успорено кретање. Ако се $a$ пише као позитиван број, за успорено кретање се пише $v = v_0 - a \cdot t$.

\section*{Средња брзина}

\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Формула & Када важи \\
\midrule
$v_{sr} = \dfrac{s}{t}$ & увек \\
$v_{sr} = \dfrac{v_0 + v}{2}$ & само код РППК \\
\bottomrule
\end{tabular}
\end{center}

Код РППК тело има средњу брзину тачно у средини временског интервала (у тренутку $\tfrac t2$).

\section*{График брзине од времена код РППК}

Код РППК брзина се мења линеарно, па је график $v(t)$ дуж; што је дуж стрмија, убрзање је веће.

\subsection*{Пример 1}
$v_0 = 0$, $a = 2\,\mathrm{m/s^2}$, $t = 4\,\mathrm{s}$.

\begin{enumerate}
  \item $v = 0 + 2\,\mathrm{m/s^2} \cdot 4\,\mathrm{s} = 8\,\mathrm{m/s}$
\end{enumerate}


\begin{center}
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\end{center}

\subsection*{Пример 2}
$v_0 = 3\,\mathrm{m/s}$, $a = 1\,\mathrm{m/s^2}$, $t = 3\,\mathrm{s}$.

\begin{enumerate}
  \item $v = 3\,\mathrm{m/s} + 1\,\mathrm{m/s^2} \cdot 3\,\mathrm{s} = 6\,\mathrm{m/s}$
\end{enumerate}


\begin{center}
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\subsection*{Пример 3}
$v_0 = 6\,\mathrm{m/s}$, $a = -2\,\mathrm{m/s^2}$, $t = 3\,\mathrm{s}$.

\begin{enumerate}
  \item $v = 6\,\mathrm{m/s} + (-2\,\mathrm{m/s^2}) \cdot 3\,\mathrm{s} = 0\,\mathrm{m/s}$
\end{enumerate}


\begin{center}
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\paragraph{Слични задаци}
\begin{enumerate}
  \item $v_0 = 0\,\mathrm{m/s}$, $a = 3\,\mathrm{m/s^2}$, $t = 5\,\mathrm{s}$. Колика је брзина на крају? Нацртај график $v(t)$.
  \item $v_0 = 2\,\mathrm{m/s}$, $a = 1{,}5\,\mathrm{m/s^2}$, $t = 4\,\mathrm{s}$. Колика је брзина на крају? Нацртај график $v(t)$.
  \item $v_0 = 10\,\mathrm{m/s}$, $a = -2\,\mathrm{m/s^2}$, $t = 5\,\mathrm{s}$. Колика је брзина на крају? Нацртај график $v(t)$.
  \item $v_0 = 12\,\mathrm{m/s}$, $a = -1{,}5\,\mathrm{m/s^2}$, $t = 4\,\mathrm{s}$. Колика је брзина на крају? Нацртај график $v(t)$.
\end{enumerate}
\noindent\textit{Решења.}

\noindent\textbf{1.} $v = v_0 + a \cdot t = 0\,\mathrm{m/s} + 3\,\mathrm{m/s^2} \cdot 5\,\mathrm{s}$; \textbf{Решење:} $v = 15\,\mathrm{m/s}$
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\noindent\textbf{2.} $v = v_0 + a \cdot t = 2\,\mathrm{m/s} + 1{,}5\,\mathrm{m/s^2} \cdot 4\,\mathrm{s}$; \textbf{Решење:} $v = 8\,\mathrm{m/s}$
\begin{center}
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\noindent\textbf{3.} $v = v_0 + a \cdot t = 10\,\mathrm{m/s} + (-2)\,\mathrm{m/s^2} \cdot 5\,\mathrm{s}$; \textbf{Решење:} $v = 0\,\mathrm{m/s}$
\begin{center}
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\noindent\textbf{4.} $v = v_0 + a \cdot t = 12\,\mathrm{m/s} + (-1{,}5)\,\mathrm{m/s^2} \cdot 4\,\mathrm{s}$; \textbf{Решење:} $v = 6\,\mathrm{m/s}$
\begin{center}
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\subsection*{Задатак}
На графику је брзина тела: $A(0\,\mathrm{s}, 4\,\mathrm{m/s})$, $B(4, 16)$, $C(6, 16)$, $D(10, 0)$. а) Опиши кретање. б) Нацртај график убрзања.

\begin{enumerate}
  \item $AB$: равномерно убрзано, $v_0 = 4\,\mathrm{m/s}$, $v_1 = 16\,\mathrm{m/s}$, $t_1 = 4\,\mathrm{s}$; $\ a_1 = \tfrac{16 - 4}{4} = 3\,\mathrm{m/s^2}$.
  \item $BC$: равномерно, $v = 16\,\mathrm{m/s}$, $t_2 = 2\,\mathrm{s}$; $\ a_2 = 0$.
  \item $CD$: равномерно успорено, од $16\,\mathrm{m/s}$ до $0$ за $t_3 = 4\,\mathrm{s}$; $\ a_3 = \tfrac{0 - 16}{4} = -4\,\mathrm{m/s^2}$.
\end{enumerate}


\begin{center}
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\begin{center}\small Дати график брзине.\end{center}

\begin{center}
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\begin{center}\small Решење под б): график убрзања.\end{center}

\paragraph{Слични задаци}
\begin{enumerate}
  \item График брзине пролази кроз тачке $A(0, 0)$, $B(3, 9)$, $C(7, 9)$, $D(10, 0)$ (време у $\mathrm s$, брзина у $\mathrm{m/s}$). а) Опиши кретање. б) Нацртај график убрзања.
\begin{center}
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\end{center}
  \item График брзине пролази кроз тачке $A(0, 2)$, $B(2, 8)$, $C(6, 8)$, $D(8, 4)$ (време у $\mathrm s$, брзина у $\mathrm{m/s}$). а) Опиши кретање. б) Нацртај график убрзања.
\begin{center}
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  \foreach \pt in {A,B,C,D} \fill (\pt) circle (1.4pt);
  \node[above] at (A) {$A$};
  \node[above] at (B) {$B$};
  \node[above] at (C) {$C$};
  \node[above] at (D) {$D$};
\end{tikzpicture}
\end{center}
  \item График брзине пролази кроз тачке $A(0, 10)$, $B(5, 0)$, $C(8, 6)$ (време у $\mathrm s$, брзина у $\mathrm{m/s}$). а) Опиши кретање. б) Нацртај график убрзања.
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \coordinate (A) at (0,162.5);
  \coordinate (B) at (163.75,0);
  \coordinate (C) at (262,97.5);
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (163.75,-3) -- (163.75,3);
  \node[below] at ($(163.75,0)!0!(163.75,0)$) {$5$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$8$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$10$};
  \draw (-3,97.5) -- (3,97.5);
  \node[left] at ($(0,97.5)!0!(0,97.5)$) {$6$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (262,97.5) -- (262,0);
  \draw[dashed] (262,97.5) -- (0,97.5);
  \draw[thick, red!70!black] (0,162.5) -- (163.75,0);
  \draw[thick, red!70!black] (163.75,0) -- (262,97.5);
  \foreach \pt in {A,B,C} \fill (\pt) circle (1.4pt);
  \node[above] at (A) {$A$};
  \node[above] at (B) {$B$};
  \node[above] at (C) {$C$};
\end{tikzpicture}
\end{center}
\end{enumerate}
\noindent\textit{Решења.}

\noindent\textbf{1.} $AB$: равномерно убрзано, од $0\,\mathrm{m/s}$ до $9\,\mathrm{m/s}$ за $3\,\mathrm{s}$; $\ a_1 = \dfrac{9 - 0}{3} = 3\,\mathrm{m/s^2}$; $BC$: равномерно, од $9\,\mathrm{m/s}$ до $9\,\mathrm{m/s}$ за $4\,\mathrm{s}$; $\ a_2 = \dfrac{9 - 9}{4} = 0\,\mathrm{m/s^2}$; $CD$: равномерно успорено, од $9\,\mathrm{m/s}$ до $0\,\mathrm{m/s}$ за $3\,\mathrm{s}$; $\ a_3 = \dfrac{0 - 9}{3} = -3\,\mathrm{m/s^2}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-97) -- (0,101);
  \draw (78.6,-3) -- (78.6,3);
  \node[below] at ($(78.6,0)!0!(78.6,0)$) {$3$};
  \draw (183.4,-3) -- (183.4,3);
  \node[below] at ($(183.4,0)!0!(183.4,0)$) {$7$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$10$};
  \draw (-3,60.67) -- (3,60.67);
  \node[left] at ($(0,60.67)!0!(0,60.67)$) {$3$};
  \draw (-3,-60.67) -- (3,-60.67);
  \node[left] at ($(0,-60.67)!0!(0,-60.67)$) {$-3$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,101)!0!(0,101)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,60.67) -- (78.6,60.67);
  \draw[thick, red!70!black] (78.6,0) -- (183.4,0);
  \draw[thick, red!70!black] (183.4,-60.67) -- (262,-60.67);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{2.} $AB$: равномерно убрзано, од $2\,\mathrm{m/s}$ до $8\,\mathrm{m/s}$ за $2\,\mathrm{s}$; $\ a_1 = \dfrac{8 - 2}{2} = 3\,\mathrm{m/s^2}$; $BC$: равномерно, од $8\,\mathrm{m/s}$ до $8\,\mathrm{m/s}$ за $4\,\mathrm{s}$; $\ a_2 = \dfrac{8 - 8}{4} = 0\,\mathrm{m/s^2}$; $CD$: равномерно успорено, од $8\,\mathrm{m/s}$ до $4\,\mathrm{m/s}$ за $2\,\mathrm{s}$; $\ a_3 = \dfrac{4 - 8}{2} = -2\,\mathrm{m/s^2}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-84.87) -- (0,113.13);
  \draw (65.5,-3) -- (65.5,3);
  \node[below] at ($(65.5,0)!0!(65.5,0)$) {$2$};
  \draw (196.5,-3) -- (196.5,3);
  \node[below] at ($(196.5,0)!0!(196.5,0)$) {$6$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$8$};
  \draw (-3,72.8) -- (3,72.8);
  \node[left] at ($(0,72.8)!0!(0,72.8)$) {$3$};
  \draw (-3,-48.53) -- (3,-48.53);
  \node[left] at ($(0,-48.53)!0!(0,-48.53)$) {$-2$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,113.13)!0!(0,113.13)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,72.8) -- (65.5,72.8);
  \draw[thick, red!70!black] (65.5,0) -- (196.5,0);
  \draw[thick, red!70!black] (196.5,-48.53) -- (262,-48.53);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{3.} $AB$: равномерно успорено, од $10\,\mathrm{m/s}$ до $0\,\mathrm{m/s}$ за $5\,\mathrm{s}$; $\ a_1 = \dfrac{0 - 10}{5} = -2\,\mathrm{m/s^2}$; $BC$: равномерно убрзано, од $0\,\mathrm{m/s}$ до $6\,\mathrm{m/s}$ за $3\,\mathrm{s}$; $\ a_2 = \dfrac{6 - 0}{3} = 2\,\mathrm{m/s^2}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-97) -- (0,101);
  \draw (163.75,-3) -- (163.75,3);
  \node[below] at ($(163.75,0)!0!(163.75,0)$) {$5$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$8$};
  \draw (-3,-60.67) -- (3,-60.67);
  \node[left] at ($(0,-60.67)!0!(0,-60.67)$) {$-2$};
  \draw (-3,60.67) -- (3,60.67);
  \node[left] at ($(0,60.67)!0!(0,60.67)$) {$2$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,101)!0!(0,101)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,-60.67) -- (163.75,-60.67);
  \draw[thick, red!70!black] (163.75,60.67) -- (262,60.67);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}


\subsection*{Задатак 2}
Тело има почетну брзину $v_0 = 28\,\mathrm{m/s}$. Првих $5\,\mathrm{s}$ убрзање је $0$, наредних $10\,\mathrm{s}$ је $-2\,\mathrm{m/s^2}$, а последњих $10\,\mathrm{s}$ је $1{,}5\,\mathrm{m/s^2}$. Нацртај график брзине од времена.

\begin{enumerate}
  \item Од $0$ до $5\,\mathrm{s}$: $v = 28\,\mathrm{m/s}$ (равномерно).
  \item После $15\,\mathrm{s}$: $v_1 = 28\,\mathrm{m/s} - 2\,\mathrm{m/s^2} \cdot 10\,\mathrm{s} = 8\,\mathrm{m/s}$.
  \item После $25\,\mathrm{s}$: $v_2 = 8\,\mathrm{m/s} + 1{,}5\,\mathrm{m/s^2} \cdot 10\,\mathrm{s} = 23\,\mathrm{m/s}$.
\end{enumerate}


\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (52.4,-3) -- (52.4,3);
  \node[below] at ($(52.4,0)!0!(52.4,0)$) {$5$};
  \draw (157.2,-3) -- (157.2,3);
  \node[below] at ($(157.2,0)!0!(157.2,0)$) {$15$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$25$};
  \draw (-3,48.53) -- (3,48.53);
  \node[left] at ($(0,48.53)!0!(0,48.53)$) {$8$};
  \draw (-3,139.53) -- (3,139.53);
  \node[left] at ($(0,139.53)!0!(0,139.53)$) {$23$};
  \draw (-3,169.87) -- (3,169.87);
  \node[left] at ($(0,169.87)!0!(0,169.87)$) {$28$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (157.2,48.53) -- (157.2,0);
  \draw[dashed] (157.2,48.53) -- (0,48.53);
  \draw[dashed] (262,139.53) -- (262,0);
  \draw[dashed] (262,139.53) -- (0,139.53);
  \draw[dashed] (52.4,169.87) -- (52.4,0);
  \draw[dashed] (52.4,169.87) -- (0,169.87);
  \draw[thick, red!70!black] (0,169.87) -- (52.4,169.87);
  \draw[thick, red!70!black] (52.4,169.87) -- (157.2,48.53);
  \draw[thick, red!70!black] (157.2,48.53) -- (262,139.53);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\begin{quote}\small\textbf{Исправка.} У свесци је у овом задатку график брзине нацртан као хоризонталне дужи (на $18$ и $13\,\mathrm{m/s}$). Код сталног убрзања различитог од нуле брзина се мења линеарно, па график мора бити коса дуж; тачан график је изнад. Рачун $v_1 = 8\,\mathrm{m/s}$, $v_2 = 23\,\mathrm{m/s}$ из свеске је тачан.\end{quote}

\paragraph{Слични задаци}
\begin{enumerate}
  \item Тело има почетну брзину $v_0 = 0\,\mathrm{m/s}$; првих $2\,\mathrm{s}$ убрзање је $5\,\mathrm{m/s^2}$, наредних $4\,\mathrm{s}$ убрзање је $0\,\mathrm{m/s^2}$, последњих $6\,\mathrm{s}$ убрзање је $-1\,\mathrm{m/s^2}$. Нацртај график брзине од времена.
  \item Тело има почетну брзину $v_0 = 20\,\mathrm{m/s}$; првих $3\,\mathrm{s}$ убрзање је $0\,\mathrm{m/s^2}$, наредних $5\,\mathrm{s}$ убрзање је $-2\,\mathrm{m/s^2}$, последњих $6\,\mathrm{s}$ убрзање је $1\,\mathrm{m/s^2}$. Нацртај график брзине од времена.
  \item Тело има почетну брзину $v_0 = 6\,\mathrm{m/s}$; првих $4\,\mathrm{s}$ убрзање је $1{,}5\,\mathrm{m/s^2}$, последњих $4\,\mathrm{s}$ убрзање је $-3\,\mathrm{m/s^2}$. Нацртај график брзине од времена.
\end{enumerate}
\noindent\textit{Решења.}

\noindent\textbf{1.} У тренутку $2\,\mathrm{s}$: $v = 0\,\mathrm{m/s} + 5\,\mathrm{m/s^2} \cdot 2\,\mathrm{s} = 10\,\mathrm{m/s}$; У тренутку $6\,\mathrm{s}$: $v = 10\,\mathrm{m/s} + 0\,\mathrm{m/s^2} \cdot 4\,\mathrm{s} = 10\,\mathrm{m/s}$; У тренутку $12\,\mathrm{s}$: $v = 10\,\mathrm{m/s} + (-1)\,\mathrm{m/s^2} \cdot 6\,\mathrm{s} = 4\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (43.67,-3) -- (43.67,3);
  \node[below] at ($(43.67,0)!0!(43.67,0)$) {$2$};
  \draw (131,-3) -- (131,3);
  \node[below] at ($(131,0)!0!(131,0)$) {$6$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$12$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$10$};
  \draw (-3,65) -- (3,65);
  \node[left] at ($(0,65)!0!(0,65)$) {$4$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (43.67,162.5) -- (43.67,0);
  \draw[dashed] (43.67,162.5) -- (0,162.5);
  \draw[dashed] (131,162.5) -- (131,0);
  \draw[dashed] (131,162.5) -- (0,162.5);
  \draw[dashed] (262,65) -- (262,0);
  \draw[dashed] (262,65) -- (0,65);
  \draw[thick, red!70!black] (0,0) -- (43.67,162.5);
  \draw[thick, red!70!black] (43.67,162.5) -- (131,162.5);
  \draw[thick, red!70!black] (131,162.5) -- (262,65);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{2.} У тренутку $3\,\mathrm{s}$: $v = 20\,\mathrm{m/s} + 0\,\mathrm{m/s^2} \cdot 3\,\mathrm{s} = 20\,\mathrm{m/s}$; У тренутку $8\,\mathrm{s}$: $v = 20\,\mathrm{m/s} + (-2)\,\mathrm{m/s^2} \cdot 5\,\mathrm{s} = 10\,\mathrm{m/s}$; У тренутку $14\,\mathrm{s}$: $v = 10\,\mathrm{m/s} + 1\,\mathrm{m/s^2} \cdot 6\,\mathrm{s} = 16\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (56.14,-3) -- (56.14,3);
  \node[below] at ($(56.14,0)!0!(56.14,0)$) {$3$};
  \draw (149.71,-3) -- (149.71,3);
  \node[below] at ($(149.71,0)!0!(149.71,0)$) {$8$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$14$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$20$};
  \draw (-3,81.25) -- (3,81.25);
  \node[left] at ($(0,81.25)!0!(0,81.25)$) {$10$};
  \draw (-3,130) -- (3,130);
  \node[left] at ($(0,130)!0!(0,130)$) {$16$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (56.14,162.5) -- (56.14,0);
  \draw[dashed] (56.14,162.5) -- (0,162.5);
  \draw[dashed] (149.71,81.25) -- (149.71,0);
  \draw[dashed] (149.71,81.25) -- (0,81.25);
  \draw[dashed] (262,130) -- (262,0);
  \draw[dashed] (262,130) -- (0,130);
  \draw[thick, red!70!black] (0,162.5) -- (56.14,162.5);
  \draw[thick, red!70!black] (56.14,162.5) -- (149.71,81.25);
  \draw[thick, red!70!black] (149.71,81.25) -- (262,130);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{3.} У тренутку $4\,\mathrm{s}$: $v = 6\,\mathrm{m/s} + 1{,}5\,\mathrm{m/s^2} \cdot 4\,\mathrm{s} = 12\,\mathrm{m/s}$; У тренутку $8\,\mathrm{s}$: $v = 12\,\mathrm{m/s} + (-3)\,\mathrm{m/s^2} \cdot 4\,\mathrm{s} = 0\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (131,-3) -- (131,3);
  \node[below] at ($(131,0)!0!(131,0)$) {$4$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$8$};
  \draw (-3,81.25) -- (3,81.25);
  \node[left] at ($(0,81.25)!0!(0,81.25)$) {$6$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$12$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (131,162.5) -- (131,0);
  \draw[dashed] (131,162.5) -- (0,162.5);
  \draw[thick, red!70!black] (0,81.25) -- (131,162.5);
  \draw[thick, red!70!black] (131,162.5) -- (262,0);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}


\subsection*{Задатак 3}
Тело има брзину $3\,\mathrm{m/s}$, а након $4\,\mathrm{s}$ брзина је $9\,\mathrm{m/s}$. Затим пет секунди иде том брзином, а три секунде успорава убрзањем $1\,\mathrm{m/s^2}$. Нацртај графике брзине и убрзања од времена.

\begin{enumerate}
  \item $a_1 = \tfrac{9 - 3}{4} = 1{,}5\,\mathrm{m/s^2}$ (од $0$ до $4\,\mathrm{s}$).
  \item $a_2 = 0$, $v = 9\,\mathrm{m/s}$ (од $4$ до $9\,\mathrm{s}$).
  \item $a_3 = -1\,\mathrm{m/s^2}$; $\ v_2 = 9\,\mathrm{m/s} - 1\,\mathrm{m/s^2} \cdot 3\,\mathrm{s} = 6\,\mathrm{m/s}$ (у $12\,\mathrm{s}$).
\end{enumerate}


\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \coordinate (A) at (0,54.6);
  \coordinate (B) at (87.33,163.8);
  \coordinate (C) at (196.5,163.8);
  \coordinate (D) at (262,109.2);
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (87.33,-3) -- (87.33,3);
  \node[below] at ($(87.33,0)!0!(87.33,0)$) {$4$};
  \draw (196.5,-3) -- (196.5,3);
  \node[below] at ($(196.5,0)!0!(196.5,0)$) {$9$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$12$};
  \draw (-3,54.6) -- (3,54.6);
  \node[left] at ($(0,54.6)!0!(0,54.6)$) {$3$};
  \draw (-3,109.2) -- (3,109.2);
  \node[left] at ($(0,109.2)!0!(0,109.2)$) {$6$};
  \draw (-3,163.8) -- (3,163.8);
  \node[left] at ($(0,163.8)!0!(0,163.8)$) {$9$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (87.33,163.8) -- (87.33,0);
  \draw[dashed] (87.33,163.8) -- (0,163.8);
  \draw[dashed] (262,109.2) -- (262,0);
  \draw[dashed] (262,109.2) -- (0,109.2);
  \draw[thick, red!70!black] (0,54.6) -- (87.33,163.8);
  \draw[thick, red!70!black] (87.33,163.8) -- (196.5,163.8);
  \draw[thick, red!70!black] (196.5,163.8) -- (262,109.2);
  \foreach \pt in {A,B,C,D} \fill (\pt) circle (1.4pt);
  \node[right] at (A) {$A$};
  \node[above] at (B) {$B$};
  \node[above] at (C) {$C$};
  \node[right] at (D) {$D$};
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-84) -- (0,114);
  \draw (87.33,-3) -- (87.33,3);
  \node[below] at ($(87.33,0)!0!(87.33,0)$) {$4$};
  \draw (196.5,-3) -- (196.5,3);
  \node[below] at ($(196.5,0)!0!(196.5,0)$) {$9$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$12$};
  \draw (-3,-52) -- (3,-52);
  \node[left] at ($(0,-52)!0!(0,-52)$) {$-1$};
  \draw (-3,78) -- (3,78);
  \node[left] at ($(0,78)!0!(0,78)$) {$1{,}5$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,114)!0!(0,114)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,78) -- (87.33,78);
  \draw[thick, red!70!black] (87.33,0) -- (196.5,0);
  \draw[thick, red!70!black] (196.5,-52) -- (262,-52);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\paragraph{Слични задаци}
\begin{enumerate}
  \item Тело има брзину $2\,\mathrm{m/s}$, а након $4\,\mathrm{s}$ брзина му је $10\,\mathrm{m/s}$. Затим $3\,\mathrm{s}$ иде том брзином, а онда $2\,\mathrm{s}$ успорава са $2\,\mathrm{m/s^2}$. Нацртај графике брзине и убрзања од времена.
  \item Тело има брзину $0\,\mathrm{m/s}$, а након $3\,\mathrm{s}$ брзина му је $12\,\mathrm{m/s}$. Затим $5\,\mathrm{s}$ иде том брзином, а онда $4\,\mathrm{s}$ успорава са $1{,}5\,\mathrm{m/s^2}$. Нацртај графике брзине и убрзања од времена.
  \item Тело има брзину $4\,\mathrm{m/s}$, а након $6\,\mathrm{s}$ брзина му је $16\,\mathrm{m/s}$. Затим $2\,\mathrm{s}$ иде том брзином, а онда $3\,\mathrm{s}$ успорава са $4\,\mathrm{m/s^2}$. Нацртај графике брзине и убрзања од времена.
\end{enumerate}
\noindent\textit{Решења.}

\noindent\textbf{1.} $a_1 = \dfrac{10 - 2}{4} = 2\,\mathrm{m/s^2}$; $a_2 = 0$, $v = 10\,\mathrm{m/s}$; $a_3 = -2\,\mathrm{m/s^2}$; $\ v_3 = 10\,\mathrm{m/s} - 2\,\mathrm{m/s^2} \cdot 2\,\mathrm{s} = 6\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \coordinate (A) at (0,32.5);
  \coordinate (B) at (116.44,162.5);
  \coordinate (C) at (203.78,162.5);
  \coordinate (D) at (262,97.5);
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (116.44,-3) -- (116.44,3);
  \node[below] at ($(116.44,0)!0!(116.44,0)$) {$4$};
  \draw (203.78,-3) -- (203.78,3);
  \node[below] at ($(203.78,0)!0!(203.78,0)$) {$7$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$9$};
  \draw (-3,32.5) -- (3,32.5);
  \node[left] at ($(0,32.5)!0!(0,32.5)$) {$2$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$10$};
  \draw (-3,97.5) -- (3,97.5);
  \node[left] at ($(0,97.5)!0!(0,97.5)$) {$6$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (116.44,162.5) -- (116.44,0);
  \draw[dashed] (116.44,162.5) -- (0,162.5);
  \draw[dashed] (203.78,162.5) -- (203.78,0);
  \draw[dashed] (203.78,162.5) -- (0,162.5);
  \draw[dashed] (262,97.5) -- (262,0);
  \draw[dashed] (262,97.5) -- (0,97.5);
  \draw[thick, red!70!black] (0,32.5) -- (116.44,162.5);
  \draw[thick, red!70!black] (116.44,162.5) -- (203.78,162.5);
  \draw[thick, red!70!black] (203.78,162.5) -- (262,97.5);
  \foreach \pt in {A,B,C,D} \fill (\pt) circle (1.4pt);
  \node[above] at (A) {$A$};
  \node[above] at (B) {$B$};
  \node[above] at (C) {$C$};
  \node[above] at (D) {$D$};
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-97) -- (0,101);
  \draw (116.44,-3) -- (116.44,3);
  \node[below] at ($(116.44,0)!0!(116.44,0)$) {$4$};
  \draw (203.78,-3) -- (203.78,3);
  \node[below] at ($(203.78,0)!0!(203.78,0)$) {$7$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$9$};
  \draw (-3,60.67) -- (3,60.67);
  \node[left] at ($(0,60.67)!0!(0,60.67)$) {$2$};
  \draw (-3,-60.67) -- (3,-60.67);
  \node[left] at ($(0,-60.67)!0!(0,-60.67)$) {$-2$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,101)!0!(0,101)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,60.67) -- (116.44,60.67);
  \draw[thick, red!70!black] (116.44,0) -- (203.78,0);
  \draw[thick, red!70!black] (203.78,-60.67) -- (262,-60.67);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{2.} $a_1 = \dfrac{12 - 0}{3} = 4\,\mathrm{m/s^2}$; $a_2 = 0$, $v = 12\,\mathrm{m/s}$; $a_3 = -1{,}5\,\mathrm{m/s^2}$; $\ v_3 = 12\,\mathrm{m/s} - 1{,}5\,\mathrm{m/s^2} \cdot 4\,\mathrm{s} = 6\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \coordinate (A) at (0,0);
  \coordinate (B) at (65.5,162.5);
  \coordinate (C) at (174.67,162.5);
  \coordinate (D) at (262,81.25);
  \draw (0,0) -- (272,0);
  \draw (0,-6) -- (0,192);
  \draw (65.5,-3) -- (65.5,3);
  \node[below] at ($(65.5,0)!0!(65.5,0)$) {$3$};
  \draw (174.67,-3) -- (174.67,3);
  \node[below] at ($(174.67,0)!0!(174.67,0)$) {$8$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$12$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$12$};
  \draw (-3,81.25) -- (3,81.25);
  \node[left] at ($(0,81.25)!0!(0,81.25)$) {$6$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (65.5,162.5) -- (65.5,0);
  \draw[dashed] (65.5,162.5) -- (0,162.5);
  \draw[dashed] (174.67,162.5) -- (174.67,0);
  \draw[dashed] (174.67,162.5) -- (0,162.5);
  \draw[dashed] (262,81.25) -- (262,0);
  \draw[dashed] (262,81.25) -- (0,81.25);
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  \node[above] at (A) {$A$};
  \node[above] at (B) {$B$};
  \node[above] at (C) {$C$};
  \node[above] at (D) {$D$};
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-69.42) -- (0,128.58);
  \draw (65.5,-3) -- (65.5,3);
  \node[below] at ($(65.5,0)!0!(65.5,0)$) {$3$};
  \draw (174.67,-3) -- (174.67,3);
  \node[below] at ($(174.67,0)!0!(174.67,0)$) {$8$};
  \draw (262,-3) -- (262,3);
  \node[below] at ($(262,0)!0!(262,0)$) {$12$};
  \draw (-3,88.24) -- (3,88.24);
  \node[left] at ($(0,88.24)!0!(0,88.24)$) {$4$};
  \draw (-3,-33.09) -- (3,-33.09);
  \node[left] at ($(0,-33.09)!0!(0,-33.09)$) {$-1{,}5$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,128.58)!0!(0,128.58)$) {$a [m/s^2]$};
  \draw[thick, red!70!black] (0,88.24) -- (65.5,88.24);
  \draw[thick, red!70!black] (65.5,0) -- (174.67,0);
  \draw[thick, red!70!black] (174.67,-33.09) -- (262,-33.09);
  \foreach \pt in {} \fill (\pt) circle (1.4pt);
\end{tikzpicture}
\end{center}

\noindent\textbf{3.} $a_1 = \dfrac{16 - 4}{6} = 2\,\mathrm{m/s^2}$; $a_2 = 0$, $v = 16\,\mathrm{m/s}$; $a_3 = -4\,\mathrm{m/s^2}$; $\ v_3 = 16\,\mathrm{m/s} - 4\,\mathrm{m/s^2} \cdot 3\,\mathrm{s} = 4\,\mathrm{m/s}$
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
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  \coordinate (B) at (142.91,162.5);
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  \draw (0,-6) -- (0,192);
  \draw (142.91,-3) -- (142.91,3);
  \node[below] at ($(142.91,0)!0!(142.91,0)$) {$6$};
  \draw (190.55,-3) -- (190.55,3);
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  \draw (262,-3) -- (262,3);
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  \draw (-3,40.62) -- (3,40.62);
  \node[left] at ($(0,40.62)!0!(0,40.62)$) {$4$};
  \draw (-3,162.5) -- (3,162.5);
  \node[left] at ($(0,162.5)!0!(0,162.5)$) {$16$};
  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
  \draw[dashed] (142.91,162.5) -- (142.91,0);
  \draw[dashed] (142.91,162.5) -- (0,162.5);
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  \draw[dashed] (190.55,162.5) -- (0,162.5);
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  \node[above] at (C) {$C$};
  \node[above] at (D) {$D$};
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[scale=0.028, every node/.style={font=\small}]
  \draw (0,0) -- (272,0);
  \draw (0,-117.22) -- (0,80.78);
  \draw (142.91,-3) -- (142.91,3);
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  \draw (190.55,-3) -- (190.55,3);
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  \draw (262,-3) -- (262,3);
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\end{center}


\end{document}