Velocity and Acceleration — Operator Notation and Parametric Equations
A free preview from The Ultimate Crash Course for STEM Majors by Jonathan David.
Operator Notation
$$
v=\frac{d}{dt}s(t)=\frac{ds}{dt}=s'(t)\equiv\dot{s}
$$
Velocity in the x-Direction
$$ v_x=\frac{d}{dt}x(t)=\frac{dx}{dt}=x'(t)\equiv\dot{x} $$Velocity in the y-Direction
$$ v_y=\frac{d}{dt}y(t)=\frac{dy}{dt}=y'(t)\equiv\dot{y} $$Note: A graph described by $y=f(x)$ can also be represented using parametric equations $x=x(t)$ and $y=y(t)$, where $y(t)=f(x(t))$. Each coordinate is expressed as a function of the parameter $t$.
Parametric Equation Graphing Example
The upper and lower halves of a circle of radius $5$ are described by:
$$ y=+\sqrt{25-x^2} \qquad\text{or}\qquad y=-\sqrt{25-x^2}, \quad -5\le x\le5 $$Together, these branches form the circle:
$$ x^2+y^2=25 $$ $$ \left(\frac{x}{5}\right)^2 +\left(\frac{y}{5}\right)^2 =1 $$Using the identity $\cos^2t+\sin^2t=1$, choose:
$$ x(t)=5\cos t, \qquad y(t)=5\sin t, \qquad t\in[0,2\pi] $$These equations trace the circle once counterclockwise, starting at $(5,0)$.
Average Acceleration — Straight-Line Motion
$$ \bar{a}\equiv a_{\mathrm{av}} =\frac{v-v_0}{t-t_0} =\frac{\Delta v}{\Delta t} $$Instantaneous Acceleration — Calculus
$$
\begin{aligned}
a
&=\lim_{\Delta t\to0}
\frac{v(t+\Delta t)-v(t)}{\Delta t}\\
&=\frac{dv}{dt}
=\dot{v}
=\frac{d}{dt}\left(\frac{ds}{dt}\right)\\
&=\frac{d^2s}{dt^2}
=\ddot{s}
\end{aligned}
$$
Acceleration in the x-Direction
$$ a_x=\frac{dv_x}{dt} =\frac{d}{dt}\left(\frac{dx}{dt}\right) =\frac{d^2x}{dt^2} =\ddot{x} $$Acceleration in the y-Direction
$$ a_y=\frac{dv_y}{dt} =\frac{d}{dt}\left(\frac{dy}{dt}\right) =\frac{d^2y}{dt^2} =\ddot{y} $$Formulas — One-Dimensional Motion
The one-dimensional motion formulas continue on the next page of the book.