HOW HIGH CAN YOU JUMP ON THE MOON? -------------------------------- John Pazmino NYSkies Astronomy Inc www.nyskies.org firstname.lastname@example.org 2008 February 16
Introduction ---------- 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.
Complications ----------- 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.
Measurements ---------- 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.
Calculation --------- The energy imparted to the jumper is the same on Moon and Earth, This is easiest treated as all potential energy doing work against gravity. 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 IN UNIFORM GRAVITY FIELD | | | | (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 ground
(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 ground.
Freeflight --------- 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 distance.
(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.
Explanation --------- 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 earth. 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.
Conclusion -------- 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 measurements. 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.