Friday, April 26, 2013

More About Friction

This item from Fair and Unbalanced knocked a stray thought loose that's been bouncing around for a while. Don't ask me how I got from Fred Flintstone to the Flash, because for the life of me I can't remember. But I do remember the basic question: what's the theoretical maximum speed a human could run?

We'll have to consider that two different ways: with and without cleats.

It's easy enough in principle to find the maximum speed of anything that has to move through the air. When the force pushing you forward is equal to the drag force exerted by the air surrounding you, you're not going to get any faster. So, we need to find the drag force:

Drag = 0.5 * (rho) * (v) * (v) * (CdS)

where rho is the density of air, v is the velocity, and CdS is the drag coefficient (Cd) times the surface area (S). It's wickedly hard to measure them for something like a running person, so we leave them lumped together. For that matter, how do we even find it? We can make an educated guess, by comparing a running dude to a falling one ... not a perfect analogy, but I know where I can look up terminal velocity. We can set the falling person's weight equal to the drag force, and then solve for CdS.

Drag = 180 lbs.
rho = 0.002377 slugs/ft^3
v = 175 ft/s

Solving for CdS gives 4.8893 ft^2.

(Incidentally: slugs are the English system's unit for mass. One slug of steel held in your hand would weigh 32.174 pounds, give or take.)

This gets us almost to the point where we can figure out the maximum speed. Now we need to figure out how a runner exerts force upon the ground. Which is why we need to split it into cases with and without cleats: with cleats, you get the added force from jabbing metal stakes into the ground, otherwise you're just relying on good old friction.

Without Cleats: The coefficient of friction between rubber and dry concrete is 0.85. This means that a 180-pound runner can only exert 153 pounds of force sideways upon the ground before he will begin to skid. Now, that lets us set up the drag equation again, and solve for v:

Drag = 153 lbs.
rho = 0.002377 slugs/ft^3
CdS = 4.8893 ft^2

Solving for v gives us 162.26 ft/s, or 110.63 miles per hour.

With Cleats: Here, the limiting factor isn't friction, it's the point where the cleats will snap off. We're going to assume "ideal" cleats here, which is to say that the cleats stick into the ground without tearing the surface. It has the dual virtue of both giving us the most beneficial possible conditions, as well as simplifying the problem. That's because now, all we have to worry about is steel's strength in shear. That's 50,000 pounds per square inch. Now, let's assume six spikes per shoe, with the spikes being 1/20 inch thick; that gives a cross-sectional area of 0.0118 square inches. In order to cause the spikes to fail in shear, you'd have to apply 589 pounds of force. This would be the maximum drag force. We can put that in the above equation, and solve for v:

Drag = 589 lbs.
rho = 0.002377 slugs/ft^3
CdS = 4.8893 ft^2

Solving for v gives us 318.38 ft/s, or 217 miles per hour.

Some interesting conclusions follow:

1) Since we know the Flash can run much, much faster than that, the obvious implication is that he's not using friction to keep his feet on the ground ... or whatever surface he's running across. (I tried to find a picture of Flash running up the side of a building. I know I've seen it. But when you Google "Flash running up the side of a building", you ... well, don't do that. Or if you do, don't come crying to me trying to unsee what you've seen.)

2) Steve Austin running 60 miles per hour? Totally possible, provided you have atomic-powered artificial legs.

3) Track and field world records have a long, long way to go. We'll probably reach human bio-mechanical limits before we even get close to the theoretical ones.

And with that, I'm gonna lace up my running shoes. I've got some work to do.

Friday, April 19, 2013

Sesquicentennial, Part XXIV: Friction


"Everything in war is simple, but the simplest thing is difficult." -- Carl von Clausewitz

"All warfare is based on deception. Hence, when we are able to attack, we must seem unable; when using our forces, we must appear inactive; when we are near, we must make the enemy believe we are far away; when far away, we must make him believe we are near." -- Sun Tzu

History tells us that the famous generals Lee and Grant first clashed in the Battle of the Wilderness, in May of 1864, but this is not entirely accurate. Lee's and Grant's decisions first began to contend with one another's in the spring of 1863, as they were each drawing up their plans of campaign for the coming season. It's not known whether either man read Clausewitz's masterwork Vom Kriege, but they were certainly becoming familiar with one of its most famous dictums. Everything they needed to do was simple, but even the simplest thing was fraught with difficulty.

Grant wanted Vicksburg. But every effort he had made to date ended in the same result: failure. He had tried twice to build canals that would get his forces in position to encircle Vicksburg, and had tried three times to use his naval superiority to bypass Vicksburg and encircle it that way. His last option was risky: march down the west bank of the Mississippi, and then cross the river below Vicksburg. Naturally, if the Confederates got wind that this was his plan, they would be able to cover the landings fairly easily. Although Grant had never read (or indeed even heard of) Sun Tzu, he nevertheless had an instinctive grasp of the obvious: he would have to deceive the Confederates of his specific intent. He would send out not one but two diversionary operations. General Sherman would command an attack on Snyder's Bluff, while Colonel Grierson would lead a sweeping cavalry raid through the entirety of central Mississippi. This, plus the thrashing around Grant had been doing for the last several months, ought to confuse the defenders of Vicksburg enough to keep them from hindering his real plan.

Oddly enough, Lieutenant General John Pemberton also wanted Vicksburg, because that was where his army was headquartered. Pemberton and Grant knew each other from their days in the Army during the Mexican War. Unusually for a Confederate officer, Pemberton was a Northerner by birth. Long years of service in the Southern states, along with having married a Virginian, shifted his sympathies such that when the break came, he went with his wife's people. In October of 1862, he was promoted to Lieutenant General and assigned the defense of Vicksburg, with orders to hold it at all costs. He didn't have nearly enough men for the task. He had 50,000 men within the works at Vicksburg, against about 100,000 men under General Grant. The only thing in his favor so far was that Grant's men were not yet at Vicksburg. This gave a small breathing space, within which he could bolster Vicksburg's defenses, provided that President Davis could conjure up the men to send him.

Sadly, conjuring would be what it would take, because there were no natural means by which President Davis could get so many soldiers.

General Joseph Johnston was in Jackson, Mississippi, and close enough to act in support, but held few men under his command. In principle, once the spring's actions in the East had been sorted out, some of Lee's troops from the Army of Northern Virginia could be detailed west to support Pemberton. But it would be up to six weeks before that decision could be taken. It would be foolish to denude Lee of men, until they knew what General Joseph Hooker would do with the Army of the Potomac.

And in any event, it might already be too late for Lee to march to Vicksburg in support.

Lee was about a thousand miles from Vicksburg. There's no practical way for an army the size of Lee's to cover that amount of ground in anything under two months. And that's by using the most direct route possible, assuming no interference by Union forces. It's safe to say that there'd be at least some effort to interfere. In any case, there are some significant dates to note: March 9, and April 24. March 9 is the last date that Lee could leave his camp and arrive at Vicksburg by May 15, and April 24 is the last day he could leave and arrive by June 30. We won't say what those last two dates signify, not quite yet. But the meaning is fairly clear, isn't it? Everyone knew that a noose was drawing around Vicksburg, the only questions were where and when.

Which brings us back to their basic problem: they still weren't entirely sure what Grant was up to. They knew he was headed to Vicksburg. But that was all they knew.

This, of course, was exactly what Grant wanted.

Saturday, April 06, 2013

Twit of the Year?

I've resisted it for some time now, but I've taken the plunge and started a Twitter account. The feed is off to the side on the right.

So far, I've found out that it's a real challenge to express a coherent thought in under 144 characters.

Anyway, if there are other feeds you think I ought to be following, let me know at @McGahaTim.