John Pazmino
 NYSkies Astronomy Inc
 2008 February 16
    I went to the 2008 February 8 meeting of the Physics Club of New 
York for a talk on astronomy database used in astrophysics and 
cosmology. After work on that Friday the 8th I hopped the train to New 
York University, where the meeting convened at 7 PM EST. I do not 
attend these meetings regularly but a couple of the club's officers 
know me from NYC Events, where they list their events. 
    By bad luck, the speaker, Dr Hogg of NYU, too sick time and had to 
pass up the meeting. The club had a substitute program, which turned 
out to be just as interesting, and more immediately useful in the 
NYSkies astronomy outreach. 
    The session became a teacher workshop on comparing jumping on 
earth with that on the Moon. Most of the PCoNY members are senior or 
retired academic folk who keep active in science education. Hence, 
this alternate presentation was warmly welcomed.
Jump mechanics
    The jump was a standing leap starting from a crouch and springing 
up. Instructions were handed out and we were divvied into teams of 
four or five persons each. My team chose up a fellow, who in my 
estimation hadn't taken a jump since trolleys quit running in New 
York. Recall that most people at the meeting were well up in years. 
This is an important factor when later I do the maths of jumping. 
    When you take a standing jump, you first crouch down to cock your 
legs, then you press quickly downward on the floor with the feet. The 
body is raised first to an upright posture, then lifted off of the 
floor for freeflight. Your body peaks in elevation, then returns to 
the ground, where your feet break the fall. 
    The body being an extended mass, we chose a point on the jumper's 
clothing near the center of mass around the hips. He wore slacks and 
belt, so we used the belt as the mark. The belt allowed the rest of us 
to observe the jump from all sides, like umpires or referees. 
    Also distributed were a tailor's tape and a strip of masking tape. 
the latter was for marking the heights of the jump phases so we could 
measure them with the tailor's tape. In our team, we did the jumps 
against an interior column in the meeting room. To further facilitate 
the work, we attached the tailor's tape to the column, zeroed at the 
floor, with small bits of masking tape. 
    Altho you cock your leg muscles in the crouch position before the 
jump it is common at the instant of springup to notch down a couple 
centimeters and them leap up. Also, when you lift off of the floor, 
your toes are extended to give the final upward kick. 
    So, we deducted two, or three?, cm from the crouch height of the 
jumper's belt and added three to the standing height to recognize this 
mechanism of the jump. 
    We did the scientific method of taking data several times and 
eyeballing the average. A curious point was that NONE of us had a 
calculette! The best we came up with was a four-banger in one team 
member's cellphone! (This WAS a physics club.) We overheard other 
teams expressing the same situation, even asking across the room for a 
calculette. With none, we rounded some numbers to ease the mental 
maths. Here I use the original figures being that while typing this 
article I got a sci/tech unit to hand. 
Physics of jump
    We assumed that the same person, of the same physical and medical 
state, jumps on Earth and Moon. We also allowed that the Moon jump is 
done indoors, like in a lunar base, so the jumper wears civvies and is 
in terrestrial air. It would not be fair to have a jump outdoors in a 
spacesuit on loose slippery lunar soil.. 
    These assumptions imply that the total energy delivered into the 
jump in the two cases is the same. The leg muscles do the same work.
    The initial phase of the jump, pushing down on the floor to 
straighten the legs and raise the body, increases the potential energy 
of the jumper. The elevation change is from the crouch height to the 
liftoff height, with the allowances noted above. 
    The actual freeflight occupies the elevation range from liftoff to 
peak. During this phase there is no force acting on the jumper from 
himself, only that from local gravity. In this phase the body has an 
initial kinetic energy, which is depleted as the jumper is slowed by 
gravity, At the peak elevation, the kinetic energy zeros out and 
acquires negative value as the body descends back to the ground. 
    At the peak the kinetic energy is fully converted to additional 
potential energy. The total potential energy, which is also the total 
energy, is that from crouch to peak elevation. 
    The total work done is equal in the two cases. What we found, and 
this is not immediately obvious, is that the proportion of 
potential/kinetic energies shifts between Earth and Moon and this 
shift is what changes the peak elevation of the jump. 
    Due to the senectude of the jumper, the numbers we captured were, 
uh, dismal. The heights were quickly marked with bits of masking tape 
on the tailor's tape and read off after each jump. With a brief huddle 
we wrote down the numbers. Several runs were done and an eyeball 
average was agreed on. 
    jump phase                      height 
    crouch, including notch-down     80 cm 
    liftoff, including toe-kick     100 cm 
    peak of freeflight              125 cm 
    The room filled with clunks and stomps as the teams did their 
runs, with some laughter. I don't think any of the jumpers did much 
better than ours. 
    The energy imparted to the jumper is the same on Moon and Earth, 
This is easiest treated as all potential energy doing work against 
    This is the ONLY 'trick' you have to know, the equation for 
potential energy in a uniform gravity field. I box it here for you. 
    |                                            | 
    | (potential energy) =                       | 
    | (mass) * (grav fld strength) * (elev chng) | 
    (potentialE) = (potentialM) 
    (massE) * (gravE) * (elevE) = (massM) * (gravM) * (elevM) 
The person has the same mass on Earth and Moon so it divides out 
    (massE) = (massM) 
    (gravE) * (elevE) = (gravM) * (elevM) 
    The gravity field strength of the Moon is quite 1/6 (this is 
rounded) that of Earth 
    (gravE) = (6) * (gravM) 
    (6)*(gravM) * (elevE) = (gravM) * (elevM) 
            (6) * (elevE) = (elevM) 
    The elevation change for a jump on the Moon is six times that on 
Earth. This is the usual answer to the question 'how high can you jump 
on the Moon?'. But not really so. Lo here the diagram and the 
figuring after it. 
Diagram of jump 
    The figure here clarifies the mechanics of the jump 
          125 cm-------------- peak height, greater on 
                  /|\    /|\                Moon than Earth  
       freeflight--|      | 
                   |      | 
                  \|/     | 
          100 cm ------------ liftoff height, same on Earth 
                   /|\    |                   and Moon 
                    |     |-- elevation change 
         springup --|     | 
                    |     | 
                   \|/   \|/ 
            80 cm ----------- crouch height, same on Earth 
                   /|\                       and Moon 
  start above gnd --| 
           0 cm ========== ground level  
Continuing ... 
    In our team's case, we got 
    (elevE) = (peakE) - (crouchE) 
            = (125 cm) - (80 cm) 
            = 45 cm 
I did tell you that the performance of our jumper was mediocre. 
    Adding back the crouch height, we get the peak height from the 
    (peakE) = (crouchE( + (elevE) 
            = ((80 cm) + (45 cm) 
            = 125 cm 
This at first seems pedantic, but we need this step on the Moon. 
    the jump on the Moon is 
    (elevM) = (6) * (elveE) 
            = (6) * (45 cm)                                  
            = 270 cm
PLUS the crouch height of 80 cm! This is the same on Moon and Earth. 
Most explanations miss this. 
    (crouchM) = (crouchE) 
    tpeakM) = (elevM) + (crouchM) 
             = (270 cm) + (80 cm)
             = 350 cm 
 .  Our jumper on the Moon could reach the ceiling of the meeting 
room. And conk his head against it. The room got a high ceiling of 
about 4 meters. 
    Note well that by this measure, the Moon jump is far LESS than 6 
times the Earth jump, measured peak to peak. 
    (peakM/E) = (peakM) / (peakE)
              = (350 cm) / (125 cm) 
              = 2.8 
    This is the number needed for, say, shooting basketballs or 
catching baseballs in sports. The higher off of the ground you can 
jump, the batter advantage you got over shooting or catching from the 
    Part of the change of elevation is the straightening of the body 
from the crouch to the liftoff positions. This is not really the 
'jump'. It's that of getting up from looking at a low bookshelf and is 
the same amount for both situations. We must remove this springup 
height from the elevation change. We need just the freeflight 
    (flightE) = (peakE) = (liftoffE) 
              = (125 cm) - (100 cm) 
              = 25 cm 
We do the same for the Moon jump 
    (flightM) = (peakM) = (liftoffM) 
              = (350 cm) - (100 cm) 
              = 250 cm 
Ratio of jumps 
    We arrive at a surprising result. The freeflight distance of the 
jumps is NOT in ratio 6:1. 
    (flightN/E) = (flightM) / (flightE) 
                = (250 cm) / (25 cm) 
                = 10.0 
    When watching a jump, you see the jumper crouch, then spring up. 
On earth and Moon this these phases look the same. The freeflight is 
much greater on the Moon, making the jumper seem superhuman. The 
jumper with the same effort shoots up faster and rises farther. The 
freeflight, even with our senior jumper, is ten times that on Earth. 
    While the total energy put into the jump is the same for both 
cases, on the Moon more of it goes into boosting the body off of the 
ground. During the springup phase, there is less downward gravity 
opposing the force upward from the legs. You 'weigh' less on the Moon. 
    At liftoff there is more energy left to give you the kinetic 
energy for freeflight. This accelerates you upward faster than on 
    Once in freeflight, the downward deceleration of gravity retards 
you slower than on Earth. You remain in the air (you're jumping 
indoors, remember?) longer and rise higher from the ground. These two 
factor combine to give you a much higher freeflight distance than 
possible on Earth. 
    The factors to mind are the elevation base level. it is NOT ground 
level but the starting height of the jump. In our experiment it's the 
crouch height marked by the jumper's belt. The other is to consider th 
freeflight distance, which is similar to the flight of a rocket. The 
motor has to lift the rocket from the ground and hurl it up to mission 
speed. If the rocket is launched from a rail or silo that's the 
equivalent of the springup phase of the jump. 
    Nothing more than high school science and algebra are needed, 
making this exercise a capital program for almost any audience and not 
just for scientists. The equations in this piece are written in BASIC 
code due to typographical limitations. They should be intelligible by 
anyone working with computers and fancier calculettes. 
    It REALLY helps to have a couple calculettes to lend out! A simple 
arithmetic unit will do for there are only the main four math 
operations in the equations. 
    The maths should be done on a worksheet with boxes to drop in the 
numbers. The instructions we had were narratives with only the margins 
to scribble the maths into.
    The measurements can be done with a meterstick but for just about 
any jump it is too short. An adult stands at belt level quite one 
meter from the ground to start with. You could use a carpenter's 
ruler, which is two or three meters long when stretched out. 
    The tailor's tape is perfect. Many can be coiled up in your kit 
for carrying, These come in two-meter length for general cloth 
    Don't forget the masking tape! It is VERY handy for ALL science 
workshops and demonstrations. Have a roll ON you when going to any of 
your science meetings. Make SURE to remove and clean up ALL tape after 
the session to leave walls and columns in good condition when you 
leave the meeting room.