MovementHow quickly did the Greek phalanx or a Roman legion move across a battlefield?I frankly have no solid idea, but I'm going to make the case that it was likely (much) slower than naively expected. So let's start with a few numbers. A sprinter at full speed can cover 10 m/s, which theoretically would translate to 600 m/min, but it is rather unlikely that this top speed can be sustained for long. Besides, a typical soldier carries armour, weapons and possibly also other gear which makes running at top speed rather hard and generally exhausting.
Hiking speedsA hiker with a fairly brisk pace can easily do 5 km per hour which translates to about 82 m/min - so that seems quite doable for a unit at first glance. However, even that is possibly on the fast side, because the hiker needs a good trail - cross-country hiking speed is a bit lower. We can not assume that the typical battlefield was well-mowed lawn - there may have been boulders, shrubcover, tufts of grass, low places covered with slippery mud or sand,... Take the following picture of the Spartan defensive position at Thermopylae for example:
But in fact usually we don't just want to move a unit across the battlefield somehow - we want it to arrive in proper formation, neatly sorted in rank and file and not as an unorganized mass of soldiers. And that means that the soldier with the most uneven footing determines the overall speed. For the parameters in the defined units, I have therefore assumed that something like a brisk hiking speed is rather rushing across the battlefield, and that normal advances (possibly under fire) are done at half that speed (which would cover a mere 30 m or so per minute, but allows time to keep shields nicely locked). It is possible to move faster and even jog across the field, but that affects both stamina of the soldiers and order of the formation negatively (however, running the last few dozens of meters for a charge seems realistic). Cavalry can move a lot faster (and is not that much affected by difficult terrain). A charging warhorse in gallop can do 45 km/h which means some 750 m/min, and even at a trott (which horses can maintain for a long time) some 200 m/min are possible - and since cavalry doesn't have the usual advantages that come from assuming a very tight formation anyway, mounted units do not need to worry about breaking formation too much.
TurningTurning around a military unit that is in any kind of formation is not easy. The reason is that to do so, every soldier needs to march at a different speed so that the whole formation wheels around - soldiers at the outside of the turn need to march quite quickly while those on the inside have to slow down or even remain near stationary. Even when this is practiced, a unit may not be able to do this successfully in the field and under attack. The Battle of Delium shows an example where the Athenian phalanx failed to do something with a victory on the flank and instead got into disarray - which even led to 'friendly fire' incidents.Turning the whole formation around when currently fighting is even more difficult, as there really is no space to perform the maneuver, soldiers at the contact line are busy fighting and can't participate in the maneuver and generally confusion and shouting level is a lot higher. The assumption is that when not fighting, units can turn some 180 degrees in a within a minute (that seems to be roughly what happens on parade grounds). When currently under attack, such a maneuver takes then several minutes (and may not even be completed before the battle is over). From this it can be understood that flank or rear attacks are extremely powerful to gain an advantage over well-arrayed heavy infantry.
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