1d Motion

Frames of Reference

Inertial Reference Frame

A frame of reference (i.e. set of coordinates) that is not accelerating.

Any frame moving at a constant relative velocity to the original is also an inertial frame.

Galilean Transformations

Position Velocity $\left (dx/dt \right)$ Acceleration $\left (d^2x/dt^2 \right)$
$$ \begin{align*} x &= x' + Vt' \\ y &= y' \\ z &= z' \end{align*} $$ $$ \begin{align*} v_x &= v'_x + V \\ v_y &= v'_y \\ v_z &= v'_z \end{align*} $$ $$ \begin{align*} a_x &= a'_x \\ a_y &= a'_y \\ a_z &= a'_z \end{align*} $$

Acceleration is the same in both frames: Galilean Invariant

What about non-inertial reference frames?

Particles in Motion

1d Equation of Motion for Constant Acceleration

\begin{equation} x = x_0 + v_0 t + \frac{1}{2}a t^2 \end{equation}

Where did this come from?

But what about when acceleration is not just a constant value?

What's the easiest $a(t)$ you can think of, where $$\frac{da}{dt} \neq 0$$

Newton's Laws

In words:

  1. No change in velocity if there is no net force acting.
  2. A force will cause a proportional acceleration inversely proportional to the mass.
  3. Forces come in equal and opposite pairs

In symbols:

  1. $\Delta v = 0 $ if $F_\textrm{net} = 0$
  2. $\mathbf{F} = m\mathbf{a}$
  3. $F_{AB} = -F_{BA}$

\begin{equation} \mathbf{F} = m\mathbf{a} \label{eq:newtons2ndlaw} \end{equation}

Or perhaps as: \begin{equation} \mathbf{F} = m \frac{d\mathbf{v}}{dt} = \frac{d\mathbf{p}}{dt} \label{eq:newtons2ndlawP} \end{equation} assuming mass doesn't change w.r.t time.

A familiar example

An object falls from a height $h$. How long does it take to hit the ground?

What is $g$ at the height of say, the International Space Station?

Drag and Air Resistance

$$\Sigma F = F(v) + \ldots = m a$$

This leads to equations of motion that resemble: $$ m \frac{dv}{dt} = -A v^n $$ or (ignoring the constants and stuff) $$ \dot{v} = -v$$

($\dot{v} \equiv \frac{dv}{dt}$ and $\ddot{v} \equiv \frac{d^2v}{dt^2}$)

Easy diff. eq:

If $$\dot{v} = v$$ what is $v(t)$?

What is $x(t)$?

Linear Drag - Bacterium

Find the $v(t)$ for a bacterium coasting in a viscous fluid. You can assume it starts with a velocity $v_0$ and then is slowed down by a force given by $F = -bv$ where $b$ is a constant that depends on the size and shape of the bacterium and the viscosity of the fluid.

Forms of Resistance

$$f(v) = b v + c v^2 = f_\textrm{lin} + f_\textrm{quad}$$

Stokes Drag

\begin{equation} F_\textrm{Stokes} = 6 \pi \eta R v \end{equation}

  • $\eta$ is the viscosity of the fluid.
  • $R$ is the radius of the object (assumed to be a sphere)
  • $v$ is the velocity of the object relative to the fluid

Newtonian Drag

\begin{equation} F_\textrm{Newtonian} = \frac{1}{2} \rho v^2 c_d A \end{equation}

  • $\rho$ is the mass density of the fluid.
  • $A$ is the cross sectional area of the object
  • $v$ is the velocity of the object relative to the fluid
  • $c_d$ is the drag coefficient

Drag Coefficients for various shapes

Reynold's number

The ratio of these two forces, the quadratic and the linear drags, can be used to describe systems: $$\mathbf{Re}\; \textrm{or}\; R \propto \frac{f_\textrm{quad}}{f_\textrm{lin}}$$

In air...

\begin{equation} \frac{f_\textrm{quad}}{f_\textrm{lin}} = \frac{cv^2}{bv} = \left(3.2 \times 10^{3} \frac{\textrm{s}}{\textrm{m}^2} \right) R v \end{equation}

We can take this ratio and use it to figure out which terms are important. All the constant are given by the various parameters of air (for example) and can be merged into one number multiplied by the radius and the speed.

History of Free Fall

What comes to mind when you see this?

Quadratic velocity dependence

Non-Linear Drag

A ball drops from a height $h$ and reaches the ground a time $t$ later. Let the ball have a mass of $m$ and a radius of $r$. What is the height of the building in terms of $t$? Air resistance is proportional the square of the velocity.

Quadratic Drag Graphs

    m = 1;
    g = 9.80;
    Cdrag = 0.5;
    \[Rho] = 1.29;
    r = 0.1;
    A = \[Pi] r^2;
    vterm = Sqrt[(2 m g )/(Cdrag \[Rho] A )];
    yDrag[t_] := vterm^2/g Log[Cosh[(g t)/vterm]]
    vDrag[t_] := vterm Tanh[(g t)/vterm]
    Plot[yDrag[t], {t, 0, 10}]
    Plot[vDrag[t], {t, 0, 10}]