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\begin{multicols}{3}
{\huge Physics}\hfill Andrew Lorimer\hspace{2em}

\section{Motion}
  \subsection*{Unit conversion}
  $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$

  \subsection*{Inclined planes}
  $F = m g \sin\theta - F_{frict} = m a$

  \subsection*{Banked tracks}
  \includegraphics[height=4cm]{/mnt/andrew/graphics/banked-track.png}
  $\theta = \tan^{-1} {{v^2} \over rg}$ (also for objects on string)

  $\Sigma F$ always acts towards centre, but not necessarily horizontally

  $\Sigma F = {{mv^2} \over r} = mg \tan \theta$

  Design speed $v = \sqrt{gr\tan\theta}$

  \subsection*{Work and energy}
  $W=Fx=\Delta \Sigma E$ (work)

  $E_K = {1 \over 2}mv^2$ (kinetic)

  $E_G = mgh$ (potential)

  $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)

  \subsection*{Horizontal motion}

  $v = {{2 \pi r} \over T}$

  $f = {1 \over T}, \quad T = {1 \over f}$

  $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$

  $\Sigma F$ towards centre, $v$ tangential

  $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$

  \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png}

  \subsection*{Vertical circular motion}
  $T =$ tension, e.g. circular pendulum

  $T+mg = {{mv^2}\over r}$ at highest point
  $T-mg = {{mv^2} \over r}$ at lowest point

  \subsection*{Projectile motion}
  \begin{itemize}
  \item{horizontal component of velocity is constant if no air resistance}

  \item{vertical component affected by gravity: $a_y = -g$}
\end{itemize}

$v=\sqrt{v^2_x + v^2_y}$ (vector addition)

$h={{u^2\sin \theta ^2}\over 2g}$ (max height)

$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight)

$d={v^2 \over g}sin \theta$ (horizontal range)
  \includegraphics[height=4cm]{/mnt/andrew/graphics/projectile-motion.png}

  \subsection*{Pulley-mass system}

  $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended

  \subsection*{Graphs}
  \begin{itemize}
    \item{Force-time: $A=\Delta \rho$}
    \item{Force-disp: $A=W$}
    \item{Force-ext: $m=k,\quad A=E_{spr}$}
  \end{itemize}

  \subsection*{Hooke's law}

  $F=-kx$

  $E_{elastic} = {1 \over 2}kx^2$

  \subsection*{Motion equations}


\begin{tabular}{ l r }
  $v=u+at$ & $x$ \\
  $x = {1 \over 2}(v+u)t$ & $a$ \\
  $x=ut+{1 \over 2}at^2$ & $v$ \\
  $x=vt-{1 \over 2}at^2$ & $u$ \\
  $v^2=u^2+2ax$ & $t$ \\
\end{tabular}

\subsection*{Momentum}

$\rho = mv$

$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$

Momentum is conserved.

$\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic

\section{Relativity}







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