EXPERIMENT $1$GRAPHICAL ANALYSISIntroductionThis experiment is designed to provide you with some basic background in the use of graphs for analyzing experimental data and results. It is important to obtain a good understanding of graphical analysis since most of the subsequent experiments you will be doing require graphs as a means of extracting results. In preparation for this experiment, you should also study the reference section on graphs in the Yellow Pages of the manual $($pages viii $-$ Xx$).$The immediate object of this laboratory exercise is to demonstrate how graphical methods can be used to determine the relationship $($formula$)$ between two physical quantities. Suppose the following measurements were obtained for the velocity of an object at various distances from the origin. An inspection of the data shows that the velocity increases as the distance increases, but says very little about the mathematical relation governing the two quantities.Table $1.$ Motion of an experimental car.$(1)$Distanced$\S$d$=0\mathrm{.}1[$m$]0\mathrm{.}61\mathrm{.}02\mathrm{.}04\mathrm{.}06\mathrm{.}08\mathrm{.}010\mathrm{.}012\mathrm{.}014\mathrm{.}016\mathrm{.}0(2)$VelocityvSv $=+0\mathrm{.}3[$m$/$s$]6\mathrm{.}16\mathrm{.}88\mathrm{.}410\mathrm{.}713\mathrm{.}014\mathrm{.}715\mathrm{.}917\mathrm{.}318\mathrm{.}919\mathrm{.}9($Note: you need to update Table $1.$ with columns containing the values of the derived variable$($s$)($d$,$ v$.$ S$($v$\backslash$deg $).$ S$($d$\text{'})$ etc.$)$ used in plotting graph$($s$)$ and performing analysis that you intend to include in the lab report$)$ProcedureThe simplest relationship that you could hope for is one where velocity y varies linearly with distance d$.$ To test whether this type of relation is valid, plot a graph of v as a function of d $($called $"$y versus d$\text{'},$ that is with v on the vertical axis and d on the horizontal axis$).$ Also indicate the uncertainty in each data$-$point by drawing error bars corresponding to $8$d and Sv$.$ Note that if the uncertainty in d is too small to show on the graph its error bar can be omitted.Can you realistically draw a straight line passing through all of the error bars on the graph? If not, the relationship is non$-$linear and you should draw a smooth curve through the error bars. In either case, it is important that you construct the "best$-$fit" line or "best$-$fit" curve such that the random scatter of data$-$points above and below the line tend to cancel each other.If the graph of $"$y versus d$"$ is linear, then your analysis is completed by simply finding the equation using the slope and intercept of the line. If the graph is not linear, your next step is to guess what new choice of variables might be linearly related and graphically test whether this is correct.For a start, you may guess that $"$y versus d$2"$ or $"$y$?$ versus $"$ are good possibilities $-$ particularly if the curvature of the $"$v vs$.$ d$\text{'}$ graph appears parabolic in nature. Compile values of the new variables $($d $3)$ and $($v$)$ as well as their respective uncertainties into your original table. $($The rules for calculating uncertainties are given in the Yellow Pages: pages xxvii $-$ xxix. Construct graphs of $"$y versus d$2"$ and $"?$ versus d$."$ If either of these graphs yields a straight line within the range of uncertainty, your search for the unknown relation is successful. The usual graphical technique of finding the slope and intercept would provide the final mathematical relation. On the other hand, if both graphs yield curves, you must continue investigating other possible relations $($e$.$g$.,$ y vs$.$ d$3,$ w$\backslash$deg vs$.$ d$,$ etc.$)$ or ask your lab instructor for advice.Analysis$1.$On the basis of your graphs, which pair of variables suggests the linear relationship governing v and d$?2.$If any one of your graphs is a straight line, find the slope and intercept of the "best$-$fit" line$\mathrm{.}3.$Also determine the possible uncertainty in the slope and intercept using the method of maximum and minimum lines $($see the Yellow Pages$).$ Compare these with the value obtained with the LINEST function in excel$\mathrm{.}4.$On the basis of your graphs and results, write your proposal for the mathematical equation relating velocity v and distance d$.($Show your reasoning$).$ Finally, check whether your equation holds by substituting some values forand d$.$Questions$1.$What are the units of the slope and intercept of your straight$-$line graph? Does this give you any clues as to what the slope and intercept might represent physically?$2.$Does it make sense to obtain straight$-$line graphs for both $"$v vs$.$ d$2"$ and $"$y$?$ vs$.$ d$"3.$What are some uses and advantages of graphs that you perceived upon doing this experiment? Are there some things which you may not have been able to determine from the table of data alone?ConclusionsIt is an important part of any experiment to make some general conclusions about the outcome and significance of the experimental investigations.What has been found. learned, or demonstrated?What does it signify or suggest?How does it correlate with expected behaviour $($if known$)?$If there is some unusual behaviour or differing results, what are possible reasons to explain this departure?