Subsection 3.1.2 Equations of Motion
The equations of motion relate the displacement of an object with its velocity, acceleration and time. The motion of a particle can follow many different paths, but here we will discuss the motion in a straight line, i.e. in one dimension. We are driving three equations of motion when acceleration is constant. From eqn. (3.1.4)
\begin{equation*}
a=\frac{\,dv}{\,dt}\quad \Rightarrow \quad \,dv=a\,dt
\end{equation*}
\begin{equation*}
\text{or,} \quad \int\limits_{v_{o}}^{v_{f}}\,dv=\int\limits_{0}^{t} a\,dt
\end{equation*}
\begin{equation*}
\text{or,} \quad \left.v\right\vert_{v_{o}}^{v_{f}}=a \left.t\right\vert_{0}^{t}
\end{equation*}
\begin{equation*}
\Rightarrow v_{f} - v_{o}=a t \quad [\because a =constant]
\end{equation*}
\begin{equation}
\therefore \quad v_{f}= v_{o}+at \tag{3.1.5}
\end{equation}
Again,
\begin{equation*}
a = \frac{\,dv}{\,dt} = \frac{\,d^{2}x}{\,dt^{2}} =\frac{\,d}{\,dt}\left(\frac{\,dx}{\,dt}\right)
\end{equation*}
\begin{equation*}
\text{or,}\quad \,d\left(\frac{\,dx}{\,dt}\right) = a\,dt
\end{equation*}
\begin{equation*}
\text{or,}\quad\int \,d\left(\frac{\,dx}{\,dt}\right) = \int a\,dt
\end{equation*}
\begin{equation*}
\text{or,}\quad \frac{\,dx}{\,dt} = a\,t + c \quad [\text{here c is a constant.]}
\end{equation*}
Now at \(t = 0\text{,}\) \(\frac{\,dx}{\,dt} = c= v_{o}\text{,}\) the initial velocity.
\begin{equation*}
\therefore \frac{\,dx}{\,dt} = a\,t + v_{o}
\end{equation*}
\begin{equation*}
\text{or,} \quad \,dx = (a\,t + v_{o})\,dt
\end{equation*}
\begin{equation*}
\text{or,} \quad \int\limits_{x_{o}}^{x_{f}} \,dx = \int\limits_{0}^{t} v_{o}\,dt +\int\limits_{0}^{t} a\,t\,dt
\end{equation*}
\begin{equation*}
\text{or,} \quad \left.x\right\vert_{x_{o}}^{x_{f}} = v_{o}\left.t\right\vert_{0}^{t} + a\left.t\right\vert_{0}^{t}
\end{equation*}
\begin{equation}
\therefore x_{f}=x_{o}+v_{o}t+\frac{1}{2}at^{2} \tag{3.1.6}
\end{equation}
Again,
\begin{equation*}
a = \frac{\,dv}{\,dt} = \frac{\,dv}{\,ds}\frac{\,ds}{\,dt} = v\frac{\,dv}{\,ds}
\end{equation*}
\begin{equation*}
\text{or,} \quad a\,ds = v\,dv
\end{equation*}
\begin{equation*}
\text{or,} \quad \int\limits_{0}^{s} a\,ds = \int\limits_{v_{o}}^{v_{f}} v\,dv
\end{equation*}
\begin{equation*}
\text{or,}\quad a \left[s\right]_{0}^{s} = \left.\frac{v^{2}}{2}\right\vert_{v_{o}}^{v_{f}}
\end{equation*}
\begin{equation}
\therefore \quad v^{2}_{f}=v^{2}_{o}+2as \qquad [here \quad s=x_{f}-x_{o}]\tag{3.1.7}
\end{equation}
From eqn. (3.1.6) if velocity is constant, then \(s = v_o t = vt \) and for free fall: \(a = g\text{,}\) (the acceleration due to gravity).
