Равномерно кретање

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\begin{document}

\begin{center}
  {\large\bfseries Кинематика} \\[2pt]
  {\small Равномерно кретање}
\end{center}

\section*{Подела према брзини}

\noindent\textbf{Равномерно кретање.} тело прелази исте дужине пута за исто време, односно брзина се током времена не мења.


\noindent\textbf{Променљиво кретање.} брзина се током времена мења.


\section*{График брзине од времена}

Код равномерног кретања график $v(t)$ је хоризонтална дуж. Пример: $v = 3\,\mathrm{m/s}$ током $t = 5\,\mathrm{s}$.

\begin{center}
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  \node[right] at ($(272,0)!0!(272,0)$) {$t [s]$};
  \node[right] at ($(0,192)!0!(0,192)$) {$v [m/s]$};
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Пређени пут је површина испод графика: $s = v \cdot t = 3\,\mathrm{m/s} \cdot 5\,\mathrm{s} = 15\,\mathrm{m}$.

\subsection*{Пример}
Тело се креће $2\,\mathrm{s}$ брзином $4\,\mathrm{m/s}$, затим $4\,\mathrm{s}$ брзином $1\,\mathrm{m/s}$, па $3\,\mathrm{s}$ мирује и на крају $1\,\mathrm{s}$ иде брзином $2\,\mathrm{m/s}$. Нацртај график $v(t)$ и израчунај средњу брзину.

\begin{enumerate}
  \item $s_1 = 2\,\mathrm{s} \cdot 4\,\mathrm{m/s} = 8\,\mathrm{m}$, $\ s_2 = 4\,\mathrm{s} \cdot 1\,\mathrm{m/s} = 4\,\mathrm{m}$, $\ s_3 = 0\,\mathrm{m}$, $\ s_4 = 1\,\mathrm{s} \cdot 2\,\mathrm{m/s} = 2\,\mathrm{m}$.
  \item $v_{sr} = \dfrac{s_1+s_2+s_3+s_4}{t_1+t_2+t_3+t_4} = \dfrac{14\,\mathrm{m}}{10\,\mathrm{s}}$.
\end{enumerate}
\noindent\textbf{Решење:} $v_{sr} = 1{,}4\,\mathrm{m/s}$


\begin{center}
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\paragraph{Слични задаци}
\begin{enumerate}
  \item Тело $3\,\mathrm{s}$ се креће брзином $5\,\mathrm{m/s}$, затим $2\,\mathrm{s}$ мирује, затим $5\,\mathrm{s}$ иде брзином $2\,\mathrm{m/s}$. Нацртај график $v(t)$ и израчунај средњу брзину.
  \item Тело $4\,\mathrm{s}$ се креће брзином $3\,\mathrm{m/s}$, затим $2\,\mathrm{s}$ иде брзином $6\,\mathrm{m/s}$, затим $2\,\mathrm{s}$ мирује, затим $2\,\mathrm{s}$ иде брзином $4\,\mathrm{m/s}$. Нацртај график $v(t)$ и израчунај средњу брзину.
  \item Тело $1\,\mathrm{s}$ се креће брзином $10\,\mathrm{m/s}$, затим $3\,\mathrm{s}$ иде брзином $2\,\mathrm{m/s}$, затим $4\,\mathrm{s}$ мирује. Нацртај график $v(t)$ и израчунај средњу брзину.
  \item Тело $6\,\mathrm{s}$ се креће брзином $5\,\mathrm{m/s}$, затим $2\,\mathrm{s}$ мирује, затим $2\,\mathrm{s}$ иде брзином $10\,\mathrm{m/s}$. Нацртај график $v(t)$ и израчунај средњу брзину.
\end{enumerate}
\noindent\textit{Решења.}

\noindent\textbf{1.} $s_1 = 3\,\mathrm{s} \cdot 5\,\mathrm{m/s} = 15\,\mathrm{m}$, $s_2 = 2\,\mathrm{s} \cdot 0\,\mathrm{m/s} = 0\,\mathrm{m}$, $s_3 = 5\,\mathrm{s} \cdot 2\,\mathrm{m/s} = 10\,\mathrm{m}$; $v_{sr} = \dfrac{s}{t} = \dfrac{25\,\mathrm{m}}{10\,\mathrm{s}}$; \textbf{Решење:} $v_{sr} = 2{,}5\,\mathrm{m/s}$
\begin{center}
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\noindent\textbf{2.} $s_1 = 4\,\mathrm{s} \cdot 3\,\mathrm{m/s} = 12\,\mathrm{m}$, $s_2 = 2\,\mathrm{s} \cdot 6\,\mathrm{m/s} = 12\,\mathrm{m}$, $s_3 = 2\,\mathrm{s} \cdot 0\,\mathrm{m/s} = 0\,\mathrm{m}$, $s_4 = 2\,\mathrm{s} \cdot 4\,\mathrm{m/s} = 8\,\mathrm{m}$; $v_{sr} = \dfrac{s}{t} = \dfrac{32\,\mathrm{m}}{10\,\mathrm{s}}$; \textbf{Решење:} $v_{sr} = 3{,}2\,\mathrm{m/s}$
\begin{center}
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\noindent\textbf{3.} $s_1 = 1\,\mathrm{s} \cdot 10\,\mathrm{m/s} = 10\,\mathrm{m}$, $s_2 = 3\,\mathrm{s} \cdot 2\,\mathrm{m/s} = 6\,\mathrm{m}$, $s_3 = 4\,\mathrm{s} \cdot 0\,\mathrm{m/s} = 0\,\mathrm{m}$; $v_{sr} = \dfrac{s}{t} = \dfrac{16\,\mathrm{m}}{8\,\mathrm{s}}$; \textbf{Решење:} $v_{sr} = 2\,\mathrm{m/s}$
\begin{center}
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\noindent\textbf{4.} $s_1 = 6\,\mathrm{s} \cdot 5\,\mathrm{m/s} = 30\,\mathrm{m}$, $s_2 = 2\,\mathrm{s} \cdot 0\,\mathrm{m/s} = 0\,\mathrm{m}$, $s_3 = 2\,\mathrm{s} \cdot 10\,\mathrm{m/s} = 20\,\mathrm{m}$; $v_{sr} = \dfrac{s}{t} = \dfrac{50\,\mathrm{m}}{10\,\mathrm{s}}$; \textbf{Решење:} $v_{sr} = 5\,\mathrm{m/s}$
\begin{center}
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\section*{График пута од времена}

Код равномерног кретања пут расте једнако за сваку секунду, па је график $s(t)$ права кроз почетак. За $v = 3\,\mathrm{m/s}$:

\begin{center}
\begin{tabular}{@{}llllll@{}}
\toprule
$t\ [\mathrm{s}]$ & 0 & 1 & 2 & 3 & 4 \\
\midrule
$s\ [\mathrm{m}]$ & 0 & 3 & 6 & 9 & 12 \\
\bottomrule
\end{tabular}
\end{center}

\begin{center}
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Два тела за исто време $t$: оно са стрмијим графиком пређе дужи пут, па има већу брзину. Ако је $s_{II} > s_I$ за исто $t$, онда је $v_{II} > v_I$.

\begin{center}
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\end{document}