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Say superheroes in a certain world have somewhat less overpowered abilities. A strong super might be able to lift a few tons, but not stop speeding trains or pick up an entire semi. Unlike the Flash, a "speedster" in this world cannot run faster than the speed of light. What speed would a person have to be running that they (their whole body not just their hand, etc.) would become hard for the eye to track/become "blurred" to someone watching them? How fast would they need to be able to accelerate to reach this point within a second of beginning to move?

Edit: Okay, I don't want to be able to turn Speedster Bob invisible through speed, just make him "look" unnaturally fast to an observer. Some people with quick reflexes may be able to move their arm and catch a thrown object, where the watcher barely had time to process their movement. An entire body is obviously much easier to see/watch and displaces more air, but how to achieve a similar effect?

Right, in response to feedback, it seems Bob cannot be made fully blurry due to his size without moving faster than the fastest vehicles known to man. Unless he's a pixie or a knome, he's out of luck. So, let's alter the question a bit and tone Bob down a bit: If Bob's top running speed is 100mph, and he can accelerate to 50 mph in less than a second and increases his acceleration until he hits this point, what would Bob look like to the observer a) when he first begins to move and b) when he hits top speed? If it makes a difference whether the observer is stationary or in motion, let's assume they are stationary at the moment.

A slowish bullet is just about visible from behind the gun, in bright sun against a dark background.

The same bullet, crossing your field of view, is invisible unless your eye happens to move at just the right time in a direction that (accidentally) tracks the bullet.

Of course, the bigger the object (like a human) the easier it is to see, at least as a blur, but the minimum that will do this for an object as much bigger than a bullet as a baseball would be close to the speed of sound. Even that would still be visible from near the path of motion (either approaching or receding), just because the angular speed is so much smaller a fraction of the actual speed in that geometry.

For something the size of a human, then, to be invisible even to someone he's running toward, he'd have to move so fast that he goes from "out of sight" in the distance to contact or passing before the brain could interpret what the eyes started to record -- say, a hundredth of a second from the horizon (or from cover) to the observer.

That would be several times the speed of sound, which would have its own effects (broken windows, deafened bystanders, burnt clothing on the speedster, etc.).

It's impossible. The human eye can track very fast objects. I mean 4 times the speed of sound fast. And distance and size makes a big difference.

For example, if you know where to look, the space station is clearly visible. Despite moving at 17,130 mph.

If the observer is focusing, there is no speed less than invisible that a person would disappear.

By the time Bob is moving fast enough to be blurry, he is breaking the sound barrier and has other things to worry about.

There are two things you need to be aware of:

Saccade

A saccade is a fast movement of an eye, head or other part of the body or of a device. It can also be a fast shift in frequency of an emitted signal or other quick change. Saccades are quick, simultaneous movements of both eyes in the same direction. Saccades are the fastest movements produced by the human body.

The peak angular speed of the eye during a saccade reaches up to 900°/s in humans! Saccades to an unexpected stimulus normally take about 200 milliseconds (ms) to initiate, and then last from about 20–200 ms, depending on their amplitude. The amplitude of a saccade is the angular distance the eye travels during the movement. For amplitudes up to about 60°, the velocity of a saccade linearly depends on the amplitude (the so-called saccadic main sequence).

In saccades larger than 60°, the peak velocity starts to plateau (nonlinearly) toward the maximum velocity attainable by the eye. For instance, a 10° amplitude is associated with a velocity of 300°/s, and 30° is associated with 500°/s. (Source_1, Source_2)

So, 0.2 seconds to initiate, 0.02 seconds best-case. Anything that changes position in less time than 0.22 seconds can't be tracked by the human eye.

Frame Rate

I'm not even going to bother quoting anyone or listing citations. So many people are comparing apples to oranges on this issue that it's hard to boil down useful citations.

Here's the argument: if your LCD screen is 2,000 pixels wide and a single pixel is moved from one side to the other at the best light-and-decay rate the screen can produce, can the human eye see it?

Yup, it can, and that can correspond to thousands of "frames per second."

What most people ignore is that it's just one dot, you're focused on it, and the actual "velocity" of transit is very slow. For example: if black-to-color-to-black is 1ms, you're only talking about 2,000ms or 2 seconds. If your screen is 20" wide, that's only 0.06 mph, which is walking speed. If it's 0.01ms, then it's 6 mph or very high-quality sprint-running speed. Do you see my point?

So, when we talk "frame rate" of the human eye, we need to talk about how fast the entire picture can change (or something major within the picture) and the eye still see it? That brings us back to Saccade movement, because it's really a question of how fast the brain-eye combination can (a) notice that something needs to be tracked, (b) lock onto that object, and (c) move the eye to maintain focus.

Conclusion

Remember that 0.02 seconds? That's the fastest (give or take, no two people are the same) the eye/head combination can track something. The 0.2 seconds is the time needed to notice and lock on. Your superheros need to be out of the frame in that time (otherwise the eye's peripheral vision is still tracking them).

This depends on how far away they are, which you didn't mention. The closer they are, the slower they can move and be out of frame before the eye can catch up. So, for illustration purposes, let's pick numbers.

Let's use the above reference's 60° number for the maximum Saccade speed. If they're 100 yards away (and simplifying this to a triangle calculation rather than an arc calculation, which would maximize the distance, but it'll be good enough), then they need to travel 100 yards in anything less than 0.22 seconds.

That's anything more than 1,363.64 feet/s²

And if you're firing a .22 caliber rimfire rifle, that really is faster than a speeding bullet.

And, like any good bullet, you didn't see him move. He was just gone.¹

Edit: Let me add a bit of context. Once, while living in Texas, I had the privilege of watching the Shuttle descend toward landing. It was (literally) like watching God draw a line of fire in the sky. It was moving a whole lot faster then 1363fps - but the size of the corridor of burning atmosphere and the distances involved were so great that I could comfortably enjoy a once-in-a-lifetime spectacle. Which is a lovely way of saying the necessary speed is completely situation-dependent.

_{¹ To be fair, the biggest reason you can't see a moving bullet is its size. Humans can see "tracer bullets" because the (I believe it is) burning phosphorous creates a much larger visible target (light glare). In a nutshell, the bullet is easier to see. Your superhero is huge, which means my analysis might be wrong because I'm not taking into account the effect of the size of the target object on visibility.}

It is not a question of speed, but rather time and situation.

The human eye blurs any actions occurring quicker than 1/50th of a second, this is why the electrical grids are set for 50-60Hz. Any quick movements done within this time period will be blurred to the eye. This includes quickly catching or throwing objects as well as any oscillatory movements such as pumping their arms while running.

With regards to full body movement, the problem that occurs is that motion is relative. While someone is unlikely to track a quick reflective or oscillatory movement (if you doubt this try any track a single blade on an aircrafts propeller at an airshow), tracking an object moving on a predictable path is much easier. If you look at a picture of a fast moving car, you will notice the background is blurred; this is due to the fact that the camera is tracking the car, if the camera was kept stationary the car would be blurred and the background would be normal. So as far a being blurred in a straight line, it is a matter of predictability and tracking of the object, speed does not matter.

Finally, another useful measure may be the minimum reaction time. Even if someone can see something, it does not mean they can react to it. The minimum time to react is around 0.15 to 0.2 seconds. In fact, the olympics actually considers it a false start if someone reacts in less than 0.1 seconds after the signal.

If you want your speedster to be blurred to a viewer over a large distance, they should zigzag at around a 50Hz frequency. If you want to just prevent someone from reacting to them, random (non-predictable) changes in direction at a 5-10Hz frequency will be enough.