Approximately how fast can people move around with the 3D Maneuver Gear (3DMG) in Attack on Titan? What is the maximum speed attainable? If people can move faster when the hook is reeled in, then I want that answer, if they can move faster in the air by swinging then I want that answer. But I don't want the speed of the hooks.
After looking through several of the episodes, looking at the initial take offs from both a stand still position and running position I have determined that the speed for take off fluctuates allot. As I am assuming this is to the amount of boost given by the user I will be working with a few different scenarios, and using rough estimates based on those scenes. As some of these scenario's
With that aside
How fast do they go on average?
During Episode 5 while Eren fights the colossal titan (3:13), he runs along the arm, attaches the hooks, and accelerates towards the giant. Using the image below as reference we can see that the distance at which he accelerates is:
roughly ~20 meters, after a ~4 meter sprint. It takes Eren ~1.5 seconds to run the distance (2.667 m/s) after which it takes him ~5.5 seconds to get behind it (3.636 m/s)
So with a running start, slightly moving upwards.
Eren can go about as fast as a bike. 8MPH (13.09 km/h)
So is this this their top speed?
Oh no, even by far from, the speeds differ greatly, and are considerably higher when moving downwards. Lets take the intro of episode 8 where Eren goes straight up, and whips himself down to the neck of a titan in roughly 0.2 seconds, over a distance of the story of 1 house which is 10 feet (3.0 m) which would already sum up at 15m/s which is already 5 times higher then the average given speed!
So would this finally be their top speed? Well, simply said. no. If for some reason they would be able to apply a whip effect to this acceleration they could reach even higher speeds. There is quite some math behind this so for exact numbers on that feel free to crack your head using these formula's but after a really quick and dirty look over it I determined it would add roughly another ~10 m/s
Adding the final speed up to ~25 m/s