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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Motion in Space : Velocity and Acceleration">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 13:32:26 1997}
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\section{Motion in Space: Velocity and Acceleration\label%
{Motion in Space : Velocity and Acceleration}}

\begin{definition}
The \textbf{velocity vector} $v(t)$ at time $t$ is defined as : 
\begin{eqnarray*}
v(t) &=&\lim_{h\rightarrow 0}\frac{r(t+h)-r(t)}{h}=r^{\prime }(t) \\
&&
\end{eqnarray*}
Thus the velocity vector points in the direction of the unit tangent vector.
\end{definition}

\begin{definition}
\label{Speed}The \textbf{\ speed }of the particle at time $t$ is the
magnitude of the velocity vector, that is, $\left| v(t)\right| .$
\end{definition}

\begin{definition}
As in the case of one-dimensional motion, the acceleration of the particle
is defined as the derivative of the velocity: 
\[
a(t)=v^{\prime }(t)=r^{\prime \prime }(t)
\]
\end{definition}

\subsubsection{Tangential and Normal Components of Acceleration\label%
{Tangential and Normal Components of Acceleration}}

It is often useful to resolve the acceleration into two components, one in
the direction of the tangent and the other in the direction of the normal.

Writing $a_{T}$ and $a_{N}$ for the tangential and normal components of
acceleration, you have 
\begin{eqnarray*}
a &=&a_{T}T+a_{N}N \\
&&
\end{eqnarray*}

Where 
\begin{eqnarray*}
a_{T} &=&v^{\prime }=\frac{v\cdot a}{v}=\frac{r^{\prime }(t)\cdot r^{\prime
\prime }(t)}{\left| r^{\prime }(t)\right| } \\
a_{N} &=&kv^{2}=\frac{\left| r^{\prime }(t)\times r^{\prime \prime
}(t)\right| }{\left| r^{\prime }(t)\right| ^{3}}\left| r^{\prime }(t)\right|
^{2}=\frac{\left| r^{\prime }(t)\times r^{\prime \prime }(t)\right| }{\left|
r^{\prime }(t)\right| ^{{}}}
\end{eqnarray*}

\subsubsection{Kepler's Laws of Planetary Motion\label%
{Kepler's laws of planetary motion}}

\begin{enumerate}
\item  A planet revolves around the sun in an elliptical orbit with the sun
at one focus.

\item  The line joining the sun to a planet sweeps out equal areas in equal
times.

\item  The square of the period of revolution of a planet is proportional to
the cube of the length of the major axis of its orbit.
\end{enumerate}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Cross product}{}{}{../../Help/DM9-1.tex#Cross product}}

{\small \hyperref{Norm}{}{}{../../Help/DM8-5.tex#Norm}}

{\small \hyperref{Vector field}{}{}{../../Help/DM9-3.tex#Vector field}}

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