Linear motion

Momentum

• $\vec{p}=m\vec{v}$ is the (linear) momentum (in $\mathsf{kg}\cdot m/s$) of a body with mass $m$ (in $\mathsf{kg}$) and velocity $\vec{v}$ (in $m/s$)

Impulse

• $\vec{\mathbf{J}}=\vec{\mathbf{F}}\Delta t$ is the impulse (in $\mathsf{N}\cdot \mathsf{s}$) delivered to an object by a constant force $\vec{\mathbf{F}}$ (in $\mathsf{N}$) over a time interval $\Delta t$ (in $\mathsf{s}$)
• $\vec{\mathbf{J}}={\int }_{t_1}^{t_2}\vec{\mathbf{F}}dt$ is the impulse delivered to an object by a variable force $\vec{\mathbf{F}}$ over the time interval $[t_1,t_2]$ (in $\mathsf{N}\cdot \mathsf{s}$)
• (Impulse-Momentum Theorem) $\vec{\mathbf{J}}=\Delta \vec{\mathbf{p}}$ (the impulse is equal to the change in momentum)

Motion in One Dimension

Constant Acceleration

• $t$ is the time duration
• $v_0$ is the initial velocity
• $x_0$ is the initial position
• $a$ is the acceleration
• $x=x_0+v_{0}t+\frac{1}{2}a{t}^2$
• $v=\frac{dx}{dt}=v_0+at$ (velocity as a function of time)
• $t=\frac{v-v_0}{a}$
• $x-x_0=v_{0}t+\frac{1}{2}a{t}^2=v_0\left[\frac{v-v_0}{a}\right]+\frac{1}{2}a{\left[\frac{v-v_0}{a}\right]}^2=\frac{v^2-v_0^2}{2a}$
• $v^2=v_0^2+2a(x-x_0)$
• $v=\sqrt{v_0^2+2a(x-x_0)}$ (velocity as a function of position)

Motion in Multiple Dimensions

todo Ballistic coefficient

% Author: Izaak Neutelings (April 2021)
\usepackage{tikz}
\usepackage{amsmath}
\usepackage{physics}
\usepackage{siunitx}
\usepackage{xcolor}
\usepackage{etoolbox} %ifthen
\usepackage[outline]{contour} % glow around text
\tikzset{>=latex} % for LaTeX arrow head
\usetikzlibrary{angles,quotes,arrows.meta} % for pic
\contourlength{1.0pt}
\colorlet{myblue}{blue!70!black}
\colorlet{mydarkblue}{blue!40!black}
\colorlet{mygreen}{green!50!black}
\colorlet{myred}{red!65!black}
\colorlet{xcol}{blue!85!black}
\colorlet{vcol}{green!70!black}
\colorlet{projcol}{vcol!90!black!60}
\tikzstyle{wave}=[myblue,thick]
\tikzstyle{xline}=[very thick,myblue]
\tikzstyle{vline}=[very thick,mygreen]
\tikzstyle{vector}=[->,very thick,vcol,line cap=round]
\tikzstyle{mydashed}=[green!30!black!90,dash pattern=on 2pt off 2pt,very thin]
\tikzstyle{mymeas}=[{Latex[length=3,width=2]}-{Latex[length=3,width=2]},thin]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.05*\ymax) --++ (#2-180:0.1*\ymax)}
 
 
\begin{document}
 
\def\xmax{3.8}
\def\ymax{2.4}
\def\v{1.0}
\def\ang{30}
\def\d{(0.9*\xmax)} % distance landing point
\def\b{tan(30)} % slope at x=0
\def\h{0.6*\ymax} % height h
\def\a{-((\b*\d+\h)/\d^2)} % coefficient
\def\nsamples{100}
 
 
 
% TRAJECTORY - PARABOLA + breakdown
\begin{tikzpicture}
  \def\v{1.4}
  \def\ang{35}
  \def\h{0.5*\ymax} % height h
  \def\vx{{\v*cos(\ang)}}
  \def\vy{{\v*sin(\ang)}}
  \coordinate (O) at (0,\h);
  \coordinate (Vx) at ({\v*cos(\ang)},\h);
  \coordinate (Vy) at (0,{\h+\v*sin(\ang)});
  \coordinate (V) at ({\v*cos(\ang)},{\h+\v*sin(\ang)});
  
  % AXES & TRAJECTORY
  \draw[->,thick]
    (-0.1*\ymax,0) -- (1.06*\xmax,0) node[right=4,below=-1] {$x$};
  \draw[->,thick]
    (0,-0.1*\ymax) -- (0,\ymax) node[below=4,left=0] {$y$};
  \draw[xline,variable=\t,samples=\nsamples,smooth,domain=0:\d+0.1]
    plot(\t,{\a*\t^2+\b*\t+\h}); %node[right=7,above=-2] {$x=x(t)$};
  
  % VELOCITY VECTOR
  \draw pic["\contour{white}{$\theta$}",draw=white,double=black,double distance=0.4,
            angle radius=13,angle eccentricity=1.4] {angle = Vx--O--V};
  \draw[mydashed]
    (Vx) |- (Vy);
  \draw[<->,projcol,thick]
    (Vy) -- (O) node[scale=0.9,midway,left=-1] {$v_{0y}$}
      -- (Vx) node[scale=0.9,midway,below=-1] {$v_{0x}$};
  \draw[->,vcol,very thick,line cap=round]
    (O) --++ ({\ang}:\v) node[above right=-4] {$\vec{v}_0$};
  \tick{O}{0} node[left] {$y_0$};
  \tick{{\d},0}{90} node[below] {$R$};
  
\end{tikzpicture}
 
 
 
\end{document}
 
 

Projectile Motion

• ${\vec{F}}_{\text{air}}=0$ (neglecting air resistance)
• ${\vec{F}}_{\text{gravity}}=m\vec{g}$ (force due to gravity)
• $\theta$ is the angle of projection
• ${\vec{v}}_0$ is the initial velocity
  • $v_{0x}=v_0\cos (\theta )$ is the initial horizontal velocity
  • $v_{0y}=v_0\sin (\theta )$ is the initial vertical velocity
• ${\vec{r}}_0$ is the initial position
  • $x_0$ is the initial horizontal position (most often $0$)
  • $y_0$ is the initial vertical position
• $\vec{v}$ is the velocity
  • $v_x=v_0\cos (\theta )$ is the horizontal velocity (constant as the initial velocity, no horizontal acceleration)
  • $v_y=v_0\sin (\theta )-gt$ is the vertical velocity
• $\vec{r}$ is the position
  • $x=x_0+v_0\cos (\theta )t$ is the horizontal position
  • $y=y_0+v_0\sin (\theta )t-\frac{1}{2}g{t}^2$ is the vertical position
• $T=\frac{2v_0\sin (\theta )}{g}$ is the time of flight (time to reach the ground)
• $y_{\text{max}}=y_0+\frac{v_0^2{\sin }^2(\theta )}{2g}$ is the maximum height
• $R=x_0+v_0\cos (\theta )T=x_0+\frac{v_0^2\sin (2\theta )}{g}$ is the range (horizontal distance)

Power

• $P=\vec{F}\cdot \vec{v}=\frac{dW}{dt}$ is the power (in $\mathsf{W}$)
  • $\vec{F}$ is the force vector (in $\mathsf{N}$)
  • $\vec{v}$ is the velocity vector (in $m/s$)