Persuading Vacationers To Calculate Latitude

peiWhen making holiday plans, golf is number 938 on my to-do list. Sure I can appreciate the energy transformations involved in swinging a club, deforming the ball and projecting it along a parabolic trajectory while dimples reduce drag. Indeed, the consistency of body mechanics and correct choice of club increase the likelihood that the ball will reach the right altitude and take the most direct path to the green. Psychologically, yes, it’s nice to control one’s emotions and play more efficiently, but I get a bigger thrill from finding someone else’s ball in the woods than from sinking a long putt. And regardless of how well I’m playing, I often harbor the thought that—even though Twain never really said it — “Golf is a good walk spoiled”.

Decades ago, I was in Prince Edward Island, Canada with some young people whose idea of a fun vacation was driving fast, smoking weed, sharing women, eating midnight snacks and sleeping past sunrise. But I figured that with the right degree of clarity and enthusiasm, I could engage them in enjoying and timing the sunset (or sunrise) from two spots on the island to calculate latitude or using the time-difference and latitude to estimate the distance between observers.

earth3The latitude is just the angle formed by 3 points: a location on earth’s surface, our planet’s center, which is the vertex, and a spot at the equator. In the diagram, an equivalent angle L is formed by the radius of the earth(Re) and the radius of the circle(r) formed at that specific latitude. A simple trigonometric ratio reveals that r = Recos L.
Prince Edward Island is not very big. Separating its easternmost point and the middle of the island is a distance of only about 80 km (50 miles). One could use a compass to make sure one point is directly west of the other. Next synchronize watches (no cell phones back then) and determine the difference in sunset times recorded by two observers 80 km apart. In temperate and tropical regions, one only needs an east-west separation distance of 75 to 110 km, respectively, to experience about a 4 minute differential.

The speed of the earth’s rotation in km/minute at that latitude is simply the circumference at the latitude divided by (24*60). This allows us to calculate the separation distance, y, between two east-west points by multiplying the earth’s rotational speed by the difference in sunset times,

latcalc1   in minutes:

But if we know the distance between two points, then the equation can be rearranged to obtain latitude:

The margins of error? At PEI’s latitude, when the observers are 80 km apart, and latcalc2= 4.0 minutes, a 6 second error would throw off the latitude by 1.6o.  An error of 2 km in judging distance would result in a 1.4 error.

I may have succeeded in getting one of my fellow-travelers to listen to my simple plan, but I suspect that once my spiel was done he said to himself, “Latitude-estimation is good weed spoiled.”
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Science Dimensions On A Baseball Diamond

Atomic and astronomical scales are not in the least intuitive. Powers of 10 tables have often been used to contrast the size of stars with the radii of atoms, but since spring training has arrived, why not use the baseball diamond to mentally animate some of the baffling dimensions of science?

dodger
 * Obviously the 1 mm proton analogy is not related to the 1 cm Earth!

If you reduce the Earth to the size of an eraser (1 cm) at the end of a pencil at home plate, the sun (109.7 times wider) becomes a 1.1 meter beach ball, 117 meters away. That would be about a meter past the left or right center field wall at Dodger Stadium. Interestingly, in this model,  the next nearest star would still be triple the distance (31 000 km) between Alaska and India. From a mass perspective, if the earth’s mass is reduced to that of a baseball, the sun would weigh as much as 644 spectators, averaging 75 kg each.

If you turn the pencil around and make a one millimeter dot on home plate to represent a proton*, well, if you scale up such a small particle to those dimensions, you will find a hydrogen  atom’s lone electron past first base, 35.3 meters away. This reveals what Rutherford demonstrated over a century ago, that an atom is mostly empty space, just like the baseball field is mostly empty if fielders are not mobile. If the mass of the electron is scaled up to that of a baseball, then the proton, which is 1836 times more massive, acquires the mass of all the bats in the clubhouse (285).

What if the Milky Way galaxy is scaled down to the size of Dodger Stadium? How small does the Earth become? Well, the Earth would only be 6 X 10-12 m wide, about 10 times smaller than the smallest atom. How small is a baseball player rendered? To about the upper limit for the size of a quark. But we’re getting too abstract. If we scale Prince Fielder’s mass down to that of an H1V virus, how massive does the Earth become? As massive as a blue whale.

 

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