How fast would a moving object have to move in order to be invisible to the naked eye? Could Dash from “The Incredibles” really run so fast we could not see him?
To see an object, light must bounce off of it and into your eye. There must be enough light that the photoreceptive neurons in the retina--the photosensitive tissue lining the back of the eye like film in a camera--are activated powerfully enough to trigger awareness. If Dash zipped through your field of view so quickly that too little light from him reached your retina, you would not see him.
What is the minimum amount of light required? In a classical experiment, flashes of light were delivered to people in complete darkness. Only the rod photoreceptors are sensitive enough to play a role at the light intensities used in this experiment. At the minimum flash brightness required to trigger visual awareness, it appeared that only a handful of rods each absorbed just one photon. The implication is striking: single cells, capturing single particles of light, can trigger perception.
Rods are most sensitive to green light (~500-nm wavelength). Dash wears red (~650 nm), and rods are only about 0.5% as sensitive to red as green. So 200-fold more light is needed to detect Dash compared with a superhero in a green costume.
However, at best, the rods can only provide a low-resolution image. To see and recognize Dash requires cone photoreceptors. Cones allow high-resolution color vision in brighter light. In today's world, where artificial light is so common, virtually all vision is cone vision. Unlike rods, individual cones must absorb very many photons to generate a sizable response. And since each cone is a single pixel in an image, many cones are required to "draw" Dash on the retina. If Dash was moving fast, photons from him would be scattered across many cones, and each of these cones might be insufficiently activated. Dash would be invisible. Or, if there was just a little more activation, he would appear as a blur.
The less light there is, the more Dash could afford to take it easy. At noon when there is a surplus of photons, he would have to be at his speediest to be invisible.
There are other factors at play. One is eye movement. We never look steadily at a single point in space. We are not aware of it, but our eyes are constantly darting from one location in the visual scene to another. The eyes make small jumps that last about a fifth of a second. These jumps are called "saccades," and our visual system shuts off during them. If Dash shot past while your eyes were making a saccade, you would not see him. If he ran past a crowd, chances are that at least some people would be in mid-saccade and totally miss him.
Another factor is attention. There are famous experiments of people being told to look for something in a video and becoming so focused on the task that they completely miss something else that should be obvious. For instance, during one of these experiments, a man in a gorilla suit walks through the video and thumps his chest. Only half the people see him! Thus, if the other Incredibles are making a diversion, Dash could take a more leisurely pace and still remain unseen.
One can continue exploring your question at length but, in fact, a complete answer does not yet exist. Precisely how much light is required for a good cone signal, how activity across the array of cones is assembled into an image, how the visual system shuts down during a saccade, and what underlies attention, are all active research areas. Your question touches on even more areas than that.
To get a sense of the scale and complexity of these research areas, click on any of the links here:
The front page looks intimidating, but most of the chapters are written with care, in an accessible style.
And nothing beats actually seeing the gorilla that people missed in that experiment on attention. You can see it in the illustrations of this paper: