Demonstrating Martian Gravity
Patrik Pirkola and Patrick B. Hall
The surface gravity on Mars is smaller than the surface gravity on Earth, resulting in longer falling times. This effect can be simulated on Earth by taking advantage of air resistance and buoyancy, which cause low density objects to %This will cause such objects to fall slowly enough to approximate objects falling on the surface of Mars. % We describe a computer simulation based on an experiment that approximates Martian gravity, and verify our numerical results by performing the experiment. % %We present a numerical recipe of such a drop. For now we call this a "Mars Simulation", although we really mean an approximation of Martian gravity. %Along with the simulation, we present an experimental verification of the result.
marsarxiv.tex
%% Modified 2005 December 5
\documentclass[12pt,preprint]{aastex}
%\documentclass[manuscript]{aastex}
%% \documentclass[preprint2]{aastex}
%% \documentclass[preprint2,longabstract]{aastex}

%% If you are submitting to a journal that translates manuscripts
%% into SGML, you need to follow certain guidelines when preparing
%% your macros. See the AASTeX v5.x Author Guide for information.
\usepackage{graphicx}
\usepackage{epstopdf}
\DeclareGraphicsExtensions{.eps}
\newcommand{\chisq}{$\chi^2$}
\newcommand{\ebv}{$E(B-V)$}
\newcommand{\ha}{H$\alpha$}
\newcommand{\hb}{H$\beta$}
\newcommand{\hi}{H\,{\sc i}}
\newcommand{\hii}{H\,{\sc ii}}
\newcommand{\hmone}{$h^{-1}$}
\newcommand{\hnone}{$h_{70}^{-1}$}
\newcommand{\ho}{$H_0$}
\newcommand{\kms}{km\,s$^{-1}$}
\newcommand{\kmsm}{km\,s$^{-1}$\,Mpc$^{-1}$}
\newcommand{\sig}{$\sigma$}

%***finish below or replace with from rs.tex

\newcommand{\alii}{\ion{Al}{2}}
\newcommand{\AlII}{\ion{Al}{2}\,$\lambda$1670}
\newcommand{\aliii}{\ion{Al}{3}}
\newcommand{\AlIII}{\ion{Al}{3}\,$\lambda\lambda$1854,1862}
\newcommand{\cii}{\ion{C}{2}}
\newcommand{\CII}{\ion{C}{2}\,$\lambda$1334}
\newcommand{\ciii}{\ion{C}{3}]}
\newcommand{\CIII}{\ion{C}{3}]\,$\lambda$1908}
\newcommand{\civ}{\ion{C}{4}}
\newcommand{\CIV}{\ion{C}{4}\,$\lambda\lambda$1548,1550}
\newcommand{\cai}{\ion{Ca}{1}}
\newcommand{\caii}{\ion{Ca}{2}}
\newcommand{\CaH}{\ion{Ca}{2}\,$\lambda$3969}
\newcommand{\CaK}{\ion{Ca}{2}\,$\lambda$3934}
\newcommand{\CaII}{\ion{Ca}{2}\,$\lambda\lambda$3934,3969}
\newcommand{\coii}{\ion{Co}{2}}
\newcommand{\crii}{\ion{Cr}{2}}
\newcommand{\FeI}{\ion{Fe}{1}\,$\lambda$2484}
\newcommand{\fei}{\ion{Fe}{1}}
\newcommand{\feii}{\ion{Fe}{2}}
\newcommand{\feiii}{\ion{Fe}{3}}
\newcommand{\feiv}{\ion{Fe}{4}}
\newcommand{\hei}{He{1}}
\newcommand{\HeI}{He{1}\,$\lambda$3188}
\newcommand{\HEI}{He{1}\,$\lambda$3889}
\newcommand{\heii}{He{2}}
\newcommand{\HeIIsf}{He{2}\,$\lambda$1640}
\newcommand{\HeIIfs}{He{2}\,$\lambda$4686}
\newcommand{\lam}{$\lambda$}
\newcommand{\lala}{$\lambda$$\lambda$}
\newcommand{\lalala}{$\lambda$$\lambda$$\lambda$}
\newcommand{\lya}{Ly$\alpha$}
\newcommand{\LyA}{Ly$\alpha$\,$\lambda$1215}
\newcommand{\lyb}{Ly$\beta$}
\newcommand{\LyB}{Ly$\beta$\,$\lambda$1026}
\newcommand{\mgi}{Mg{1}}
\newcommand{\mgiii}{Mg{3}}
\newcommand{\mgx}{Mg\,{\sc x}}
\newcommand{\MgI}{Mg{1}\,$\lambda$2852}
\newcommand{\MgII}{Mg{2}\,$\lambda\lambda$2796,2803}
\newcommand{\mni}{Mn{1}}
\newcommand{\mnii}{Mn{2}}
\newcommand{\nai}{Na{1}}
\newcommand{\NaI}{Na{1}\,$\lambda\lambda$5891,5897}
\newcommand{\Niv}{N{4}}
\newcommand{\Nv}{N{5}}
\newcommand{\NV}{N{5}\,$\lambda\lambda$1238,1242}
\newcommand{\neiii}{[Ne{3}]}
\newcommand{\NeIIIa}{[Ne{3}]\,$\lambda$3869}
\newcommand{\NeIIIb}{[Ne{3}]\,$\lambda$3969}
\newcommand{\nev}{[Ne{5}]}
\newcommand{\neviii}{Ne\,{\sc viii}}
\newcommand{\NeVa}{[Ne{5}]\,$\lambda$3346}
\newcommand{\NeVb}{[Ne{5}]\,$\lambda$3426}
\newcommand{\NIii}{Ni{2}}
\newcommand{\oi}{O{1}}
\newcommand{\OI}{O{1}\,$\lambda$1302}
\newcommand{\OII}{[O{2}]\,$\lambda\lambda$3727,3729}
\newcommand{\Oiii}{O{3}}
\newcommand{\OIII}{O{3}\,$\lambda$3133}
\newcommand{\ov}{O{5}}
\newcommand{\ovi}{O{6}}
\newcommand{\OVI}{O{6}\,$\lambda\lambda$1031,1037}
\newcommand{\SV}{S{4}\,$\lambda$1501}
\newcommand{\Si}{S{1}}
\newcommand{\SIi}{Si{1}}
\newcommand{\SIii}{Si{2}}
\newcommand{\SIiii}{Si{3}}
\newcommand{\Siiii}{Si{3}]}
\newcommand{\SIiv}{Si{4}}
\newcommand{\SIxii}{Si\,{\sc xii}}
\newcommand{\SiII}{Si{2}\,$\lambda$1263}
\newcommand{\SiIII}{Si{3}]\,$\lambda$1892}
\newcommand{\SiIV}{Si{4}\,$\lambda\lambda$1393,1402}
\newcommand{\sio}{Si{4}/O\,{\sc iv}]}
\newcommand{\SiO}{Si{4}/O\,{\sc iv}]\,$\lambda$1400}
\newcommand{\znii}{Zn{2}}

\slugcomment{Submitted to Physics Education May 19, 2015}
%\shorttitle{Collapsed Cores in Globular Clusters}
%\shortauthors{Djorgovski et al.}

\begin{document}

\title{Demonstrating Martian Gravity} %, by Hall \& Pirkola}

\author{Patrik Pirkola and Patrick B. Hall}
\affil{Department of Physics and Astronomy, York University, Toronto, Canada}
%\clearpage

\begin{abstract}

The surface gravity on Mars is smaller than the surface gravity on Earth, resulting in longer falling times.
This effect can be simulated on Earth by taking advantage of air resistance and buoyancy, which cause low density objects to %This will cause such objects to 
fall slowly enough to approximate objects falling on the surface of Mars. 
%
We describe a computer simulation based on an experiment that approximates Martian gravity, and verify our numerical results by performing the experiment.
%
%We present a numerical recipe of such a drop. For now we call this a "Mars Simulation", although we really mean an approximation of Martian gravity.
%Along with the simulation, we present an experimental verification of the result.
\end{abstract}

%\keywords{***}

\section{Introduction}


The gravitational acceleration $g$ at the surface of an approximately
spherical planet is $g=GM/R^2$, where $G$ is the gravitational constant
and $M$ and $R$ are, respectively, the mass and radius of the planet.

%The mass and radius of Mercury are both s
%Mercury has both a smaller $M$ and a smaller $R$ %than Mars,
%but in just the right proportion that its $g$ is %equal to that of Mars.

The time $T$ required to fall from a height $h$ due to only
a gravitational acceleration $g$ %, neglecting air resistance, 
can be derived from the equation for distance travelled in time $T$
at a fixed acceleration:
\begin{equation}
h = \frac{1}{2}gT^2 ~~\longrightarrow~~ T(h) = %\sqrt{\frac{2h}{g}}
\sqrt{2h/g}.
\end{equation}
When the acceleration varies (due to air resistance),
a computer simulation can be used to calculate the fall time $T$.

%For Earth, the Moon and Mars (or Mercury), 
%Table 1 gives the gravitational acceleration $g$
%as well as the time required to fall from heights %of 1, 2 and 3 meters,
%ignoring air resistance on Earth or on Mars.
We have designed an experiment on Earth to match the fall time of an object on Mars.  The MarsSim row in Table 1 gives the fall time results of a computer simulation using the parameters of that experiment.  A drop height of 2 meters matches the fall time on Mars and produces a relatively good match to Martian dynamics. 
%(The times for Mercury are almost identical to those of Mars, because
%Mercury's smaller size is compensated for by its higher density.)

%On Earth, 
\section{Logistics}
\subsection{Theory}
The net acceleration at time $t$
on a falling object of total mass $m_{\rm tot}$
is a downward (negative) gravitational acceleration 
plus upward air resistance and buoyancy terms:
%$ a(t) = -g + C[v(t)]^2 + F_b/m_{\rm tot} $
%where the coefficient $C$ and the buoyancy force $F_b$ 
%both depend in part on the geometry of the object.
\begin{equation}
a(t) = -g + \frac{1}{2}\rho_{\rm air} C_{\rm drag} A [v(t)]^2 /m_{\rm tot} + F_{\rm buoy}/m_{\rm tot}
\end{equation}
where the density of air is $\rho_{\rm air}$,
the drag coefficient of the object is $C_{\rm drag}$,
the horizontal cross-sectional area of the object is $A$,
and the object's downward velocity is $v(t)$.
The buoyancy force is $F_{\rm buoy}=m_{\rm air}g$, 
where $m_{\rm air}$ is the mass of air displaced by the object.
%Note that $m_{\rm tot} = m_{\rm obj} + m_{\rm air}$,
%where $m_{\rm obj}$ is the mass of the object itself.
\subsection{The Simulation}
For our simulation, the object is a hollow rectangular box or bag of 
vertical depth $d$, length $\ell$ and width $w$ (so that $A=\ell w$),
with sides of negligible thickness and with mass $m_{\rm obj}$.
%
For such an object, %the buoyancy force $F_{\rm buoy}=m_{\rm air}g$
the mass of air inside the box is %the density of air times the box volume, 
$m_{\rm air}=\rho_{\rm air}\ell w d$ and
the total mass of the falling box is $m_{\rm tot}=m_{\rm obj}+m_{\rm air}$.
%The air resistance coefficient is $C=\frac{1}{2}\rho_{\rm air} C_{\rm drag} \ell w/m_{\rm tot}$
%where $C_{\rm drag}$ is the drag coefficient of the object.
%
The drag coefficient for a rectangular box will lie between 
$C_{\rm drag}=1.05$ (cube, $\ell=w=d$) and $C_{\rm drag}=1.17$ 
(thin rectangular plate with $\ell/w<5$ and $d\ll\ell$; Hoerner 1965).
We assume $C_{\rm drag}=1.11$ for our rectangular box with 
$\ell\simeq w$ and $d/\ell\simeq 0.5$.
%
Using the above quantities in the expression for $a(t)$, the velocity $v(t)$ 
and height fallen $z(t)$ can be found numerically starting from
$ a(0)=-g $, $ v(0)=0 $ and $ z(0)=h $.
The time $T$ required to fall a distance $h$ is found by
stopping the calculation when $z(T)=0$.

We use an empty, rectangular, vinyl bag
(Richards Homewares Clear Vinyl Jumbo Blanket Bag No.~441W)
with $\ell=0.533$\,m, $w=0.635$\,m and $d=0.279$\,m,
so that with $\rho_{\rm air}=1.225$ kg\,m$^{-3}$, $m_{\rm air}=0.116$\,kg.
The bag itself weighs $m_{\rm obj}=0.140$\,kg.
%when removed from its storage case.
%
When spray-painted to resemble a Martian rock 
using a can of orange or reddish paint 
%(e.g., Krylon\textsuperscript{\textregistered}
%Make It Stone!\textsuperscript{\texttrademark} 
%Textured Paint, Metallic Copper \#48262),
the bag weighs $m_{\rm obj}=0.180$\,kg.
%432\,g container to start
%132\,g container at end
%yet only 40g paint came out: compressed 7.5:1
\\With $\rho_{\rm air}=1.225$ kg\,m$^{-3}$, $C_{\rm drag}=1.11$, $A$ =  0.3385 m$^{2}$, $m_{\rm tot}$ =  0.2960 kg, and $F_{\rm buoy}$ = 1.1348 N,
we can then write (2) as
\begin{equation}
a(t) = -g + b[v(t)]^2  + a_{\rm buoy}
\end{equation}
where $b$ = 0.7774 m$^{-1}$ and $a_{\rm buoy}$ = 3.834 m\,s$^{-2}$.
\section{Results}
Table 1 gives the calculated times for the spray-painted bag to fall 
distances of 1, 2 and 3 meters on Earth.  
Those times match the times required for an object to fall
from the same heights on Mars to within $\pm$11\%.
%5\%, 7.5\%, and 19\%, respectively.
The fall time from 2 meters was experimentally verified to be
$1.06 \pm 0.17$ seconds using 
3 observers' measurements of 3 separate drops. Since this drop was the best match to Martian gravity, we include its computer simulated dynamics as compared to real Martian gravity as Figure 1. We also include an image of the bag as Figure 2.
\begin{table}[!h]
  \caption{Gravitational Accelerations and Fall Times}
 % \centerline{Table 1: Gravitational Accelerations and Fall Time}
\smallskip
  \begin{tabular}{@{}lcccc@{}}
\hline
Planet & ~$g$ (m/s$^2$)~ & ~$T$~(1\,m)~ & ~$T$~(2\,m)~ & ~$T$~(3\,m)~ \\
\hline
%...	g	1m	2m	3m
Earth & 9.81 &  0.452 & 0.639 & 0.782 \\
%Venus & 8.89 &  0.474 & 0.671 & 0.822 \\
Mars*  & 3.70 &  0.735 & 1.040 & 1.274 \\ %also Mercury	0.377g
%MarsSim1 & ... & 0.700 & 1.117 & 1.518 \\ %unpainted	% 1.917
MarsSim & ... & 0.656 & 1.038 & 1.404 \\ %painted	% 1.917
Moon  & 1.63 &  1.108 & 1.567 & 1.919 \\	% 2.216	%	0.166g
\hline
\end{tabular}\\
\label{tab:1}
%\caption{Table 1}

*Also Mercury, which has nearly identical g.
\end{table}
\begin{figure}[!h]
 \includegraphics[scale=0.4]{marsim222.pdf}

 \caption{Two meter Mars simulation versus Martian drop.}
 %\centerline{Figure 1: Two meter Mars simulation versus Martian drop.}
\end{figure}
\begin{figure}[!h]
 \includegraphics[scale=0.2]{mars.png}

 \caption{The bag used for the experiment.}
 %\centerline{Figure 1: Two meter Mars simulation versus Martian drop.}
\end{figure}
\section{Conclusion}
This simulation can be used as a straightforward demonstration 
of the dynamics in Martian gravity as compared
to Earth gravity, by simultaneously dropping from the same height, an identical bag filled with textiles.

This simulation can also be used to illustrate the concept of 
air resistance.  Rotating the bag can yield different values of $A$,
resulting in different fall times.  
Furthermore, fall time measurement distributions, averages and
root-mean-square uncertainties can be calculated 
by having students time one or more drops
%(e.g., with stopwatch apps on cellular phones) 
and record their measurements.

Alternatively, this simulation can be presented as a computational
assignment.  For example, given an object of fixed volume and mass (matching 
the spray-painted bag), find the horizontal surface area it must have so that the fall time is the same on Earth as on Mars for a given height.

~

~

%\clearpage

%***To match Moon, try removing zippers from 2 bags and sewing together?

%v(1)=v(0)+dt*a(0)
%z(1)=z(0)+dt*0.5*(v(0)+v(1))
%a(1)=b+C*v(1)^2-g


%\acknowledgments


%\bibliography{/Users/phall/OLDWORK/AASTEX/pathall}
\section*{References}
\begin{enumerate} 
\item Hoerner, S. F. 1965, Fluid Dynamic Drag, 2nd edition (Bakersfield, CA: Hoerner Fluid Dynamics), 3-16
\end{enumerate}


%\clearpage




\end{document}


