When you jump into the air the height you reach is by how strong you are, how much you weight and gravity. We don't have much control over gravity without a bit of serious travel, but what if we could dial it back? Imagine travel to other worlds where the gravitational field is less - where you weigh less. The gravitational acceleration depends on the radius of the world and its mass.^{1} Relative to Earth the moon's surface gravity is about 0.17, Mercury is 0.38, Mars 0.38, and the tiny Martin moon of Phobos is only about 0.005.

My vertical jump is approximately embarrassing, but Colleen is athletic and has trained hers and got to where she could raise her center of mass 31 inches.
She does this by converting energy from her muscles into a vertical velocity and reaches her peak height when that velocity has burned off - her initial energy = m_{colleen}gh. If you solve for h you note that if g is varied, her ultimate jump height varies as the inverse. So her 31 inches on Earth becomes a bit over 15 feet on the Moon and 515 feet on Phobos! Of course her weight on Phobos is only 14 ounces.

You can also calculate how long she would be above the ground.^{2} On Earth her flight is about four tenths of a second, but on the Moon it would increase to 5.7 seconds and Phobos gives a leisurely long trip of about 2 minutes and 40 seconds.

Imagine what sports would be like (assuming you could breathe, of course)!

Phobos would present some interesting challenges as the escape velocity is about 25 miles per hour. It would be easy to throw a ball fast enough to leave the little moon and sail off into space. You can also imagine constructing some not terribly cumbersome wings and flying around in an air filled room on our moon. Any number of fantastic new sports could be created. For fun let your imagination take flight and out a few numbers so you can invent some new sports for the Moon, Mars and beyond. We appear to be wired to take delight in sport and I have no doubt this will happen some day even if it is largely impractical now.^{3}

But we can move around on the Earth and find a bit of advantage for another reason...

The Earth spins about on its axis once every 24 hours. There is a centripetal force from this spinning that tries to pull something off the surface. If you are on the equator the force of gravity plus the centripetal force makes you effectively a bit lighter. At the poles the outwards force vanishes and you are a bit heavier. A bit of physics and geometry gives the effective value for g is about 9.8061[1 - 0.00263cos(2Θ]), where Θ is your latitude in degrees.^{4} Plugging in the extremes of 0 and 90 degrees ...

you weigh about 0.53% less at the equator than at the poles

There are a few other terms - one that depends on your height about sea level - you weigh about 0.11% less at 3,000 meters than at sea level.... and yet another smaller term that depends on the tides.^{5 } The bottom line is that on a spherical rotating planet like the Earth with its variations in altitude there will be significant differences in the acceleration of a body due to gravity that depend on where you are.

If you want to lift the largest weight or jump as high as possible, you want to go as close to the poles as you can. Going to a higher altitude might hurt your athletic performance as there is less oxygen, and most Summer Olympic events seem to be held close to sea level these days.

But the real world is a bit more complex. Even if you include these corrections, very careful measurements of g show something else is lurking under the surface. The big one is the shape of the Earth. Our planet turns out to be far from spherical, but is rather a bumpy ellipsoid, so the altitude dependent terms needs correction. Reference geoids are calculated and are an important component of the GPS system. We also have high quality maps of the Moon and Mars...^{6}

Under the surface are regions that vary in density - mountains and continents tend to be lighter material that "floats". Maps of the distribution of mass under your feet changes depending where you are - accurate gravitation maps have been tabulated by experimentation with satellite observations greatly increasing the accuracy of the models. The value for g changes by about 0.2% within the boundaries of the lower 48 states. Across the planet the difference can be as large as 0.8%.

Position is something you need to correct for if you care about precision weights and measures, but it is something that sport does not consider. There are usually other factors that are larger in sports performance, so we can get away with ignoring it - but for some sports like weight lifting it may make a difference. But if you really need to lose that extra pound and are into extreme travel ....

And some day we may be doing sport in spaceflight using simulated gravity (a rotating the spacecraft for example) or even on another planet or moon....

... then things get fun!

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^{1} The force of gravity is Gm_{1}m_{2}/r^{2} where r is the distance between two bodies, m_{1} and m_{2} are their masses and G is a constant of nature. We simply this on Earth by noting that force = mass times acceleration, so for a body on the Earth's surface the force of gravity is the mass of the object times the acceleration due to gravity. A bit of algebra shows this is Gm_{earth}/r^{2}_{earth } It shows up in physics texts as g. Your weight is the force that holds you to the planet and is just this acceleration times your mass.

The important thing to note is g depends on the radius and mass of the body you are standing on.

^{2} distance = gt^{2}/2, so t = 2sqrt(2h/g) because it takes the same amount of time to go up as it does to fall. Note that this neglects air resistance.

^{3} This would be fertile ground for future completely immersive computer games where you senses can be altered - or at least a powerful enough simulation can give a sensation of very low gravitational accelerations.

In the real universe a few astronauts managed to jump around on the moon and Alan Shepard even smuggled some golf balls and his six iron on Apollo 14 managing what was probably the more impressive shots ever

^{
4} the centripetal force is mv^{2} /R_{earth} where v is the speed the Earth's surface is moving at at a given latitude. At the equator v is about 463 meters per second and it is 0 at the poles.

^{5} At sea level g = Gm_{1} /r^{2} , so at altitude h, gh = Gm_{1} /(r+h)^{2} = Gm_{1} /(r^{2} + 2rh+h^{2}). Noting h^{2}/r^{2} is extremely small and rearranging we get gh ~ g/(1+2h/r). Note that the centripetal term changes a bit as the radius changes from r to r + h. Plugging in the numbers this altitude dependent term where g at a given latitude decreases by about 3.92 *10^{-7}h, where h is the distance above sea level in meters. It is straightforward to calculate the tidal term, but I won't in interesting of time.

^{6} Very accurate gravity maps can be made by carefully tracking the position of an orbiting satellite. The best are made using a pair of orbiting satellites flying in formation perhaps a hundred kilometers apart. They compare their relative positions to better than a millimeter. Depending on what is directly under them, they speed up and slow down changing their separation a bit as they fly. One of the more impressive measurements in science.

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Recipe corner

**Barbecued Okra**

Okra has a bad reputation - largely from inappropriate cooking techniques. Getting it right is simple - just wash and carefully dry some fresh okra, string on a pair of bamboo skewers (don't run one through your finger!), brush with olive oil and a finishing salt like Maldon and throw on a hot grill for about 10 to 15 minutes turning a few times and brushing on more oil along the way.

When experimenting with fresh produce it is often best to just try simple things and explore the flavors using various techniques rather than relying on many seasonings (other than salt) and other ingredients.

**Rugbrød**

Europeans usually have much better breads than Americans. I'm a fan of the dark and dense ones - Danish rugbrød in particular. Making it isn't easy and I can't find anything like it in bakeries (I keep trying - if you have a source, please let me know!). I even wrote to the Danish embassy in Manhattan and had a reply lamenting the poor state of bread in the US. They get deliveries from the motherland, but the staffer suggested I try a Swedish mix from Ikea.

It turns out to work very well - look for Brödmix Flerkorn (Swedish for multi-grain bread mix). This is the old packaging. Ikea has largely burned through that stock and has the same thing under their own brand with directions in English. Making it is very very easy. Just add some water to the milk carton-ish container, shake and pour into a baking pan. You let the yeast do its thing for awhile and then bake. It isn't the best rugbrød I've had, but it is still very good and a great basis for open face sandwiches. Very healthy stuff too...

recommended!!

back to regular recipes soon...

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