\subsection{MC-1c~~Motion, Velocity and Acceleration}

\noindent
OBJECTIVES:
\begin{enumerate}\setlength{\itemsep}{1pt}
\item[] The major objectives of this exploratory computer lab are two-fold.  Since you will be using computer based data-acquisition throughout this course, we expect you to become familiar with the  PASCO{\small \copyright} (page~\pageref{PASCO}) interface hardware and software.   Our second objective is for you to develop an intuition for Newtonian mechanics by experimenting with 1-D motion.  With the exception of the first part, calibration of the sonic sensor, there is no extensive write-up in this lab, but only a series of recommended experiments and the requirement to write down your observations in your lab book/form. \\
 \end{enumerate}\latex{\vspace*{-.05in}}

\noindent THEORY:
\begin{enumerate}\setlength{\itemsep}{1pt}
\item[] The motion of an object is described by
indicating its distances $x_1$ and $x_2$ from a fixed reference point at two different times $t_1$ and $t_2$. From the change in position between these two times one calculates the average {\em velocity} (remember that direction is implied) for the time interval: \\ $$ \mbox{average velocity} \equiv \overline{v} = \frac{x_2 - x_1}{t_2 - t_1} \;=\; \frac{\Delta x}{\Delta t} \ m/s$$\\ The {\em acceleration} of an object is found by finding its velocity $v_1$ and $v_2$ at two different times $t_1$ and $t_2$. From the change in velocity between two different times one calculates the average acceleration for the time interval: \\ $$ \mbox{average acceleration} \equiv \overline{a} = \frac{v_2 - v_1}{t_2 - t_1} \; = \; \frac{\Delta v}{\Delta t} \; m/s^2 $$ 
\end{enumerate}\latex{\vspace*{-.05in}}


{\em  FUNDAMENTAL CONCEPTS:} 
\begin{enumerate} \setlength{\itemsep}{2pt}
\item The equation that describes the motion of an object that 
moves with {\em constant velocity} is: $ x = A + B \cdot t$. \\ 
If you make a plot of $x$ versus $t$, you find that it describes a 
straight line. The letter $A$ indicates the position of the object at time $t=0$.  The letter $B$ is the slope of the line, and is equal to the velocity of the object. So we can rewrite this $x = x_0 + v \cdot t$.

\item The equation that describes the motion of an object that 
moves with {\em constant acceleration} is: 
$x = x_0 + v_0 \cdot t + \frac{1}{2} a t^2$ and  $
v= v_0 + at$. \\
So we can rewrite this as: 
$x = A + Bt + Ct^2$  and $ v = B + 2Ct$. 
The letter $A$ indicates the position of the object at time $t=0$. 
The letter $B$ is the
the velocity of the 
object at time $t=0$, and is the  slope of the graph at this time.
The letter C is equal to half the acceleration.  
\end{enumerate}

\noindent \underline{{\bf PRECAUTIONS:}} 
\begin{enumerate}
\item[] In order for the position sensor to work properly it must be pointed in such away that it ``sees'' the vane, and doesn't identify the front of the cart; that means that it must be pointed slightly upwards. The sonic ranger tends to ``see'' the closest reflecting surface.  In addition the minimum range is approximately 40 $cm$.  \\ Make sure you do not drop the carts or allow them to roll off the table, because it damages the bearings and they begin to suffer too much friction.  Try to arrange to keep the cart on the track all the time.
\end{enumerate}\latex{\vspace*{-.10in}}

\noindent APPARATUS \latex{\vspace*{-.05in}}
\begin{enumerate}\setlength{\itemsep}{0pt}
\item[$\Rightarrow$] Computer  with monitor, keyboard and mouse.
\item[$\Rightarrow$] A PASCO position sensor; this device emits a series of short pulses of sound, and receives 
the echo of the sound reflected by a nearby object. The length of the  time interval between the emission and the reception of the sound pulse depends on the distance to the reflecting object. \\
This method of locating an object is the same as the one used by bats to find flying insects or by navy ships to locate submarines.
\item[$\Rightarrow$] A PASCO Signal Interface converts the time interval between the emission and reception of the sound pulse to digital form, i.e.,  numbers that can be then plotted on the monitor. 
\item[$\Rightarrow$] PASCO dynamic track with magnetic bumpers; cart with reflecting vane; 
meter stick; one or two steel blocks.\latex{\vspace*{-.08in}}
\end{enumerate}

\noindent{PROCEDURE:} \latex{\vspace*{-.05in}}
\begin{enumerate}\setlength{\itemsep}{0pt}
\item[]  Your instructor will demonstrate how to configure the experiment.  To initiate the PASCO(page~\pageref{PASCO}) interface software you will need to click the computer mouse on the telescope icon in the ``toolkit'' area below.  The Fig.~1 below shows how the display should appear.  Note that, while you are able to reconfigure the display parameters, the default values that are specified on start-up will allow you to do this experiment without necessitating any changes.  All three measured quantities, position, velocity, and acceleration, are displayed simultaneously.  Since velocity is determined from the position data and acceleration from the velocity the ``scatter'' in the data will become progressively more pronounced.\latex{\vspace*{-.1in}}
\input{toolbar/m01c_tb1}
\begin{figure}[!h] \centering 
%\includegraphics[width=4.0in]{figs/motion0.eps}
\includegraphics[width=5.0in]{figs/motion0r.eps}
\caption {The PASCO Data Studio display format}
\end{figure}
\end{enumerate}

\noindent{Experiment I, Basic Operation and Sonic ranger calibration:} 
\begin{enumerate} 
\item{To start the data acquisition CLICK on the START icon.  To stop it CLICK on the same stop which will now become the STOP icon. Each run gets its own data set in the ``Data'' display window. If there are any data sets in existence  you will {\em not} be able to reconfigure the interface parameters or sensor inputs, unless you clear (delete) all data.  The is also a monitor feature that can be accessed on the EXPERIMENT pull down menu (or Alt-M).}

\item{With the data acquisition started move the cart {\em to} and {\em fro} and watch the position, velocity and acceleration displays.} 

\item{STOP the data acquisition.}

\item{CLICK once (or twice) on the SCALE TO FIT icon.  CLICK on the ZOOM SELECT icon and move to one of graph regions and CLICK and DRAG the mouse.  CLICK on the CROSSHAIR icon and move about on the various graphs.  Change the plot display region be manual adjusting the $x$-min, $x$-max, etc.\ values.  To do this double CLICK within a plot window.  CLICK on the STATISTICS icon once and then again.  {\em Make} sure that all members of the group have an opportunity to test these components.  It will facilitate the rest of the lab course if 
the basic operations on the software interface are understood by everyone!}

\item{DELETE the data set by a CLICK on the RUN \#1 item in the ``Data'' window and then striking the ``Delete'' ($<$DEL$>$) key.}

\item{Start the data acquisition and observe the closest distance to the sonic ranger at  which it still functions.  This value is supposed to be close to 40 $cm$.  If it is much larger, re-aim the position sensor.}

\item{Configure the distances so that when the cart nearly touches the near magnetic bumper the sonic ranger still records accurately.}
\item{Measure the position at two distances approximately 1 $m$ apart and compare the printed centimeter scale with the position sensor readout.  By how much do the readings differ?}
\end{enumerate}




%%\end{document}
\noindent{Experiment II, Inclined Plane and Motion:} \begin{enumerate} 
\item{Raise the side of the track furthest from the position sensor using one
of the supplied blocks.  Hang the 500 $g$ weight off the other side to weigh the
front end down and prevent unwanted slippage of the track (See the demonstration
set-up to see how the weight should be positioned.) }
\item{Find an appropriate release point that allows the cart to roll down the
track {\em without} striking the magnetic bumper.}
\item{CLICK the START button and release the cart letting it bounce three or four times and the CLICK the STOP button.}
\item{Qualitatively describe the shape of the three curves: position, velocity
and acceleration and discuss how they evolve with time.}
\item{Obtain a hard copy of this data by simultaneously pressing the {\bf CTRL-p} keys.}
\item{Label/identify the various key features in the various curves by 
writing directly on the hard copies.  Paste the graphs in your note book.}
\end{enumerate}

\indent QUESTIONS: (to be discussed as a group)
\begin{enumerate}
\item{Does the velocity increase or decrease linearly with time when it is sliding up or down the track?}
\item{When the velocity is close to zero can you observe any discrepancies in the data?  Can you think of a reason for any deviations from linearity?}
\item{Why does the maximum of the position readout fall with each subsequent bounce?}
\item{Is the collision with the magnetic bumper or the residual friction in the
bearing the most likely source of loss?}
\item{Can you think of a method using your data to determine which of these two proposed mechanisms is the most likely culprit?}
 \end{enumerate} 

\noindent{Experiment III, Reproducing Expt. II manually:} 
\begin{enumerate} 
\item{Remove the block so that the cart is no longer raised.}
\item{Start a new data set (by clicking the START button) and try to move
the cart back and forth at constant velocity using your hand from the
side.  Alternatively have one team member aim the position sensor at
another member holding a book waist-high and walk towards the sensor
or away from the sensor.}
\item{Using the cross hair estimate your velocity in
either $m/s$ or $cm/s$.}
\item{Repeat step 3.\ but now try to obtain a region of constant acceleration.} 
\end{enumerate}

\noindent{Experiment IV, Acceleration at $g$ (9.8 $m/s^2$):}
\begin{enumerate}  
\item Have one member of the group stand {\em carefully} on a chair and hold 
the position sensor facing downward above another member's head while he or
she is holding a notebook on their head.
\item Start a new data set and have the student holding the notebook jump up
and down a few times. 
\item Stop the recording and determine whether free-fall yields a constant 
acceleration close to the accepted value. 
\item You may wish to try to fit data and  measure the
acceleration.  First use the ZOOM SELECT icon to choose a range of data. 
The CLICK on the FIT icon to get at the pull down menu.  You could use
the position data and use a quadratic function or the velocity data and use
a linear fit (or even the acceleration curve).  Ask you lab instructor for a demonstration
if you are unsure.
\item Fini. 
\end{enumerate}
