MotionEquations123

\documentclass{article}
\usepackage{amsmath}

\title{Motion Equations Derivation}

\begin{document}

Algebraic derivations \\

\textbf{Velocity-Time} \\
Starting with the definition of acceleration.

% Equation 1
\begin{align*}
    a &= \frac{\Delta\nu}{\Delta t} \\
    a &= \frac{\nu-\upsilon}{t} \\
    \nu &= \upsilon+at \tag{1}
\end{align*}

where, \\
a = acceleration \\
$\Delta\nu$ = change in velocity  \\
t = time \\
$\nu$ = final velocity \\
$\upsilon$ = initial velocity \\

\textbf{Position-Time} \\
% Equation A
Starting with the definition of average velocity.
\begin{align*}
    \overline{\nu} &= \frac{\Delta s}{\Delta t} \\
    \overline{\nu} &= \frac{s-s_0}{t} \\
    s &= s_0 + \overline{\nu} t \tag{A}
\end{align*}

where, \\
$\overline{\nu}$ = average velocity \\
$\Delta$ s = change in distance \\
s = final position \\
$s_0$ = initial position

Continuing with the mean speed theorem.
\begin{align*}
    \overline{\nu} &= \frac{1}{2}(\nu+\upsilon) \tag{4}
\end{align*}

Substitute (1) into (4).
\begin{align*}
    \overline{\nu} &= \frac{1}{2}[(\upsilon+at)+\upsilon] \\
    \overline{\nu} &= \frac{1}{2}(2\upsilon+at) \\
    \overline{\nu} &= \upsilon + \frac{1}{2}at \tag{b}
\end{align*}

Substitute (b) into (a) and solve for s.
\begin{align*}
    s &= s_0 + (\upsilon + \frac{1}{2}at)t \\
    s &= s_0 + \upsilon t + \frac{1}{2}at^2
\end{align*}

Also written as,
\begin{align*}
    \Delta s &= \upsilon t + \frac{1}{2}at^2 \tag{2}
\end{align*}

\textbf{Velocity-Position} \\
Starting by solving (1) for time.
\begin{align*}
    \nu &= \upsilon + at \\
    t &= \frac{\nu - \upsilon}{a}
\end{align*}

Substitute into (2),
\begin{align*}
    s &= s_0 + \upsilon t + \frac{1}{2}at^2 \\
    s &= s_0 + \upsilon
        \left (\frac{\nu - \upsilon}{a} \right)
        + \frac{1}{2}a
        \left (\frac{\nu - \upsilon}{a} \right) ^2 \\
    s - s_0 &= \frac{\nu\upsilon - \upsilon^2}{a}
        + \frac{\nu^2 - 2\nu\upsilon + \upsilon^2}{2a} \\
    2a(s - s_0) &= 2(\nu\upsilon - \upsilon^2)
        + (\nu^2 -2\nu\upsilon + \upsilon^2) \\
    2a(s - s_0) &= \nu^2 - \upsilon^2 \\
    \nu^2 &= \upsilon^2 + 2a(s - s_0)
\end{align*}

Also written as,
\begin{align*}
    \nu^2 &= \upsilon^2 + 2a(\Delta s)\tag{3}
\end{align*}

\end{document}