I've always wondered about something that I find a bit subtle. Suppose a cart is rolling and as it rolls, a heavy object drops straight down into the cart. What happens? The cart continues moving, but at a lower speed. The best way to understand this is conservation of momentum. The cart gives some of its horizontal momentum to the object, so the final velocity is between the two initial velocities.
Now, the situation is much more complicated in the treadmill situation. For one thing, the treadmill belt is being moved by a motor, and for another, the runner's foot moves past their center of mass as they run, so the belt and the runner aren't really "coupled" like in the cart example. I believe the momentum of the belt could be roughly replaced with its "inertia" (how hard it would be to stretch and/or hold in place). For the second complication, I think a runner's center of mass does move backwards while they are on the belt.
If these assumptions hold, the result would be that the belt slows down when the runner is on the ground. Then it speeds up when the runner is in the air, and so on. It's a very minor version of hanging on to the armrests, and the result would be that the treadmill would think the runner is running faster than they actually are.
Any physicists or engineers here have an opinion on this? There's got to be some belt slowdown when the runner is on the belt. Is it enough to make a significant difference in recorded speed?