omenti

controlling what you can - the art of the high jump

I while back I asked  if there was interest in a few Olympic themed posts - after all, sport is a gateway to many other areas.  Charlie replied:

 

How about the high jump? I was fascinated by Randall Cunningham's 18 year old daughter's comment on getting her head over the bar and her body would follow. Also, her body didn’t look like it had the right muscles for her accomplishments. Anything interesting there?

 

Vashti Cunningham is a world class high jump athlete. The highest levels of competition in hotly contested sports select ideal body types - while body type is important, it is required.  Vashti is currently listed at something over 6-foot-1 and weighs 121 pounds.  She probably has an extremely high proportion of fast twitch muscle and other genetic gifts, not to mention determination, and for events like this you don't need huge muscle mass as long as its the right kind of muscle but more on that later.

 

The high school physics approach to the high jump would be to recognize you can turn kinetic energy into potential energy in much the way a roller coaster works - you trade speed to fight gravity to altitude and visa versa.  Her kinetic energy depends on her mass and the square of her velocity:  E_kin = 1/2 * mv².and her potential energy depends on how high she is: E_pot = mgh where g is the acceleration due to gravity.   Set these equal and mass drops out leaving height =  v²/2g.  She is probably something over 10 meters per second as she transitions to her jump - 22 to 23 mph.  This gives around 5 meters of altitude gain - much higher than any high jump record. 

 

Fundamentally high jump is not an efficient way to turn kinetic into potential energy.  You want to stop quickly converting all of the energy of motion into throwing your body skyward.  To do that you need a machine - a lever in the form of a pole.  

 

High jumping is a pure sport - no apparatus, just see how high you can jump.   And it it has its tricks and surprises.  Vashti, like most high jumpers, makes use of the changeable shape of her body and physics.   Her motion can be described as the motion of her center of mass - the unique point around which her mass is distributed.¹  If she's standing, given her body type, Vashti's center of mass is about 55 percent of her height - a bit over a meter in her case.  In the high jump she launches her center of mass in an arc.   All work she does goes into raising her center of mass hopefully about the bar.

 

A body's center of mass is not necessarily inside of it.   Throw a horseshoe or boomerang and you see a dramatic illustration as the center of mass follows a smooth path with the body spinning around it.   You can bend your body and move your center of mass outside of it.  This allows for innovation discovered by Dick Fosbury - the Fosbury Flop.  Vashti bends her body around the bar - her center of mass clears the bar, but her body only does when it has to. ²

 

Now she's using physics properly, but there is more going on.  Her center of mass is high in the first place.  Her limbs are long and light making it easier to bend her body.  Height is generally bad for distance, but a plus for sprints.   Sprinters don't need huge muscle mass - just enough to get up to speed and launch.   The muscles in her back are probably very well developed.  She'd probably be good at yoga. 

 

Now it gets into technique.  I talked with a Division-I track coach who didn't want his name used.  Much of what she wants to do is in the run-up and takeoff. The run-up develops speed - a lot of speed - and positions the body so the foot hits the right place for takeoff to begin.  She plants her foot ahead of her body which is traveling over twenty miles per hour at this point.  Her leg is mostly firm but she's fighting its flexion.  That builds up enormous forces in her takeoff leg's muscles allowing her to create a large downward force as her forward momentum comes almost to zero in well under two tenths of a second (she's decelerating at well over six gs - peaking much higher).³

 

This phase is complex.  She's also swinging her arms upwards - hard - along with her other leg.  This pushes her core body down compressing the muscles in her takeoff leg even more.   During the run-up she want to have a low position - that big vertical push needs a build up over the takeoff phase. She's adding more swinging her non-contact limbs up and the low position at plant and high position at take-off usually means a long contact time to get as much of the muscle force to the ground as possible.   Jumpers run very fast with their hips in a low position - a few centimeters can make a big difference.

  

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¹ A bit more technically the center of mass of a distribution of mass (planet, ball, person, car, ... anything with mass) is the unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation.  Note that center of gravity is often used in place of center of mass if the acceleration due to gravity does not vary (much) over the body.

² You see this in ballet when a great dancer pulls off a flawless grand jeté.  She raises her legs into a nearly horizontal position and raises her arms above her head dropping the level of her head relative to her center of mass as it rises. She reverses the process as she begins to fall earthward again. I timed one at 0.44 seconds - a very long illusion. I also timed a basketball player in a slam dunk contest at about 0.3 seconds. In theory he could probably do it longer with his longer limbs, but he lacked her technique.  Three tenths of a second still looks startling - even beautiful to the audience.  Hang time in volleyball is more on the order of basketball - I've timed some centers at a bit over 0.3 seconds.  It must feel wonderful when properly executed.

³ For simplicity I'm avoiding talking about tendons.  They act as springs - mechanical batteries that store energy and able to release it very quickly.  Their size and stiffness make enormous differences and have a large genetic component. 

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Recipe Corner

It is high Summer and we're getting a wide variety of everything including great tomatoes!  Nothing specific, but there has been some discussion about less than perfect looking produce - the so-called ugly fruits and vegetables.  

Consumers demand cosmetically perfect examples - we seem to equate quality with visual appearance (as we do with too many things we don't understand well). Large amounts are discarded by markets and even larger amounts are simply not harvested by farmers as they don’t believe there is a market. Wastage is about 40% - driving up prices and making healthy eating more difficult as a result.

Generally ugly produce is just fine. In fact - it may even be better for you.  (the link is to NPR's The Salt)

 

notes on a pioneering data scientist

Recently I sent out a note that was well received by several people, so I reproduce it here:

 

 

There are a few friends I take long wandering conversations through a city with. Great conversation is always wonderful, but I tend to get distracted. A few years ago I a conversation with Juliette had turned to data flow. I had remembered something neat from when I was an undergrad and was getting ready to talk when Juliette saved me from being mowed down by a car. That was enough of a reset that I went blank and she moved on to something that seemed more interesting.

Somehow the lost thought returned. It stuck me that a larger group may be interested..

The Crimean Was is often remembered as a war of technological firsts … the first use of the telegraph, photography, railroads for troop and supply transport, a variety of weapon technologies and modern nursing practices (I’m sure the list is much longer). People today tend to remember the telegraph and nursing … the details of the war, who, why and how.. have faded into irrelevance for most of us. But mention nursing and Florence Nightingale comes to mind.

She was certainly a nurse, but she was also fantastic convincing the powers that be that soldiers were dying in hospitals at a greater rate than on the battlefield. It turns out she was an early data science person skilled at data collection, vetting, analysis and presentation. She was a pioneering figure of what some call data science these days. She invented some graphics techniques and (I think) coined the term applied statistics and managed to get it into the curriculum at Oxford and Cambridge.

The early 19th century saw an explosion in applied mathematics. She managed to catch the bug and took very detailed records in nursing homes, Her mathematical fame is from her rose chats - polar area diagrams. it was possible to compactly and clearly present dense information to people without deep statistical backgrounds - generals and politicians for example. She used applied math to properly focus resources…

so that’s it Juliette — Florence Nightingale’s most important contribution may have been being able to lobby and focus attention to the real problem that she could illuminate as an applied mathematician

a minute of searching produced one of her more famous charts:

 

 

a vanishing natural resource

We have our fundamental needs. I want to shiver with awe and wonder at the Universe.

 

Last week I was walking in a relatively dark area around home and found myself looking towards Sagittarius - one of the most beautiful regions of the night sky. On clear dark nights the Northern Hemisphere Summer Milky Way is particularly rich -you're looking towards the center. At the center is a supermassive black hole with about four million solar masses and a diameter about seventeen times that of the Sun.¹ There is a lot going on in the region, but it is about 26,000 light years away with a lot of dust preventing imaging it - at least in visible wavelengths.  Exotics aside - looking at this area with binoculars or just the naked eye on the right sort of night stirs something in us.

 

The problem is finding a proper night sky. Sky brightness prevents a third of the world's population from seeing the Milky Way - a number that rises to eighty percent in the US.  Living in New Jersey I have to travel for real darkness.   I was able to travel to some very dark areas just outside the city and, on moonless nights, being in the mountains would show a sky so rich with stars that finding constellations was an impossible task. The zodiacal light and gegenschein were easy and the Milky Way casts shadows.  Finding your way around was difficult - and the only things visible were mountains cutting their blackness into the starry sky.²

 

Some years ago a fellow traveler forwarded a proposal for rating seeing conditions.  It isn't as empirical as some of the scientific measurements of sky brightness and turbulence that you would use for observatory siting, but they're a good way for amateur astronomers to describe the sky:

 

Class 1: Excellent dark-sky site. The zodiacal light, gegenschein, and zodiacal band are all visible - the zodiacal light to a striking degree, and the zodiacal band spanning the entire sky. Even with direct vision, the galaxy M33 is an obvious naked-eye object. The Scorpius and Sagittarius region of the Milky Way casts obvious diffuse shadows on the ground. To the unaided eye the limiting magnitude is 7.6 to 8.0 (with effort); the presence of Jupiter or Venus in the sky seems to degrade dark adaptation.³ Airglow (a very faint, naturally occurring glow most evident within about 15 degrees of the horizon) is readily apparent. With a 32-centimeter (12½) scope, stars to magnitude 17.5 can be detected with effort, while a 50-cm (20-inch) instrument used with moderate magnification will reach 19th magnitude. If you are observing on a grass-covered field bordered by trees, your telescope, companions, and vehicle are almost totally invisible. This is an observer's Nirvana!

Class 2: Typical truly dark site. Airglow may be weakly apparent along the horizon. M33 is rather easily seen with direct vision. The summer Milky Way is highly structured to the unaided eye, and its brightest parts look like veined marble when viewed with ordinary binoculars. The zodiacal light is still bright enough to cast weak shadows just before dawn and after dusk, and its color can be seen as distinctly yellowish when compared with the blue-white of the Milky Way. Any clouds in the sky are visible only as dark holes or voids in the starry background. You can see your telescope and surroundings only vaguely, except where they project against the sky. Many of the Messier globular clusters are distinct naked-eye objects. The limiting naked-eye magnitude is as faint as 7.1 to 7.5, while a 32-cm telescope reaches to magnitude 16 or 17.

Class 3: Rural sky. Some indication of light pollution is evident along the horizon. Clouds may appear faintly illuminated in the brightest parts of the sky near the horizon but are dark overhead. The Milky Way still appears complex, and globular clusters such as M4, M5, M15, and M22 are all distinct naked-eye objects. M33 is easy to see with averted vision. The zodiacal light is striking in spring and autumn (when it extends 60 degrees above the horizon after dusk and before dawn) and its color is at least weakly indicated. Your telescope is vaguely apparent at a distance of 20 or 30 feet. The naked-eye limiting magnitude is 6.6 to 7.0, and a 32-cm reflector will reach to 16th magnitude.

Class 4: Rural/suburban transition. Fairly obvious light-pollution domes are apparent over population centers in several directions. The zodiacal light is clearly evident but doesn't even extend halfway to the zenith at the beginning or end of twilight. The Milky Way well above the horizon is still impressive but lacks all but the most obvious structure. M33 is a difficult averted-vision object and is detectable only when at an altitude higher than 50 degrees. Clouds in the direction of light-pollution sources are illuminated but only slightly so, and are still dark overhead. You can make out your telescope rather clearly at a distance. The maximum naked-eye limiting magnitude is 6.1 to 6.5, and a 32-cm reflector used with moderate magnification will reveal stars of magnitude 15.5.

Class 5: Suburban sky. Only hints of the zodiacal light are seen on the best spring and autumn nights. The Milky Way is very weak or invisible near the horizon and looks rather washed out overhead. Light sources are evident in most if not all directions. Over most or all of the sky, clouds are quite noticeably brighter than the sky itself. The naked-eye limit is around 5.6 to 6.0, and a 32-cm reflector will reach about magnitude 14.5 to 15.

Class 6: Bright suburban sky. No trace of the zodiacal light can be seen, even on the best nights. Any indications of the Milky Way are apparent only toward the zenith. The sky within 35 degrees of the horizon glows grayish white. Clouds anywhere in the sky appear fairly bright. You have no trouble seeing eyepieces and telescope accessories on an observing table. M33 is impossible to see without binoculars, and M31 is only modestly apparent to the unaided eye. The naked-eye limit is about 5.5, and a 32-cm telescope used at moderate powers will show stars at magnitude 14.0 to 14.5.

Class 7: Suburban/urban transition. The entire sky background has a vague, grayish white hue. Strong light sources are evident in all directions. The Milky Way is totally invisible or nearly so. M44 or M31 may be glimpsed with the unaided eye but are very indistinct. Clouds are brilliantly lit. Even in moderate-size telescopes, the brightest Messier objects are pale ghosts of their true selves. The naked-eye limiting magnitude is 5.0 if you really try, and a 32-cm reflector will barely reach 14th magnitude.

Class 8: City sky. The sky glows whitish gray or orangish, and you can read newspaper headlines without difficulty. M31 and M44 may be barely glimpsed by an experienced observer on good nights, and only the bright Messier objects are detectable with a modest-size telescope. Some of the stars making up the familiar constellation patterns are difficult to see or are absent entirely. The naked eye can pick out stars down to magnitude 4.5 at best, if you know just where to look, and the stellar limit for a 32-cm reflector is little better than magnitude 13.

Class 9: Inner-city sky. The entire sky is brightly lit, even at the zenith. Many stars making up familiar constellation figures are invisible, and dim constellations such as Cancer and Pisces are not seen at all. Aside from perhaps the Pleiades, no Messier objects are visible to the unaided eye. The only celestial objects that really provide pleasing telescopic views are the Moon, the planets, and a few of the brightest star clusters (if you can find them). The naked-eye limiting magnitude is 4.0 or less.

 

I've probably had a dozen nights of Class 1 (mostly above 3000 meters) and a few hundred of Class 2 - the last Class 2 was in the Arizona mountains a few years ago.⁴  Sadly Class 7 or worse dominates my local seeing. The aurora is a very rare sight and the Milky Way is impossible . There is a certain quality of life associated with real darkness - like being around frogs and insects in the Summer.

 

A few people are trying to protect the darkest areas as well as cut wasteful lighting.

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¹ The size of a black hole is usually described by its Schwarzschild radius-  the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light.  It turns out to be proportional to the black hole's mass and for simple non-rotating black holes is 2MG/c².   If the Sun could collapse to a black hole its Schwarzschild radius would be about 3 kilometers.  If the Earth could collapse it would be nine millimeters.  Very tiny black holes of lessor mass evaporate, but that is another story...

²  There are fun things you can try with your eyes when its very dark.  Here's an example - search this blog using 'dark' for a few more

³ Magnitude is a modern adaptation of an ancient measure of a star's brightness.  The Greeks thought brighter stars were physically larger and assigned a ranking - very bright first magnitude objects down to dim sixth magnitude.  It was eventually realized stars appear as point objects with differing apparent brightnesses.  A good enough approximation to the ancient assignments is the fifth root of 100 - about 2.512.  A first magnitude object is 2.512 times brighter than a second.  Arcturus is 0 and the scale goes negative for very bright objects.  The Sun is about -27, Venus sometimes gets in the -4 range and the full moon is about -13.  Professional and amateur astronomers both use the modern defintion

⁴ Amateur astronomers are attracted to dark areas and some buy land to site small private observatories.  In addition to low sky brightness and average/seasonal cloud cover, atmospheric turbulence is a big issue.  Sometimes otherwise identical properties with a kilometer of each other can vary in price by as much as a factor of five mostly on turbulence issues

 

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Recipe corner

I've been traveling and not cooking much for myself - at least not creatively.   A recent learning is Indian condiments are great on veggie burgers.