Motion blur is the streak-like effect that occurs when shooting a still image or video, because your subjects are moving rapidly through the frame, or the camera exposure is particularly long (i.e. time-lapse photography). This effect can be found in human eyes as well. If your eye moves past an object (or vice versa), the image will have a motion blur, unless you're tracking the object at the same speed, which is called "smooth pursuit".
Motion Blur in Photography
Photos are never literal — they can represent a mood, an action, a long stretch of time, or anything else the photographer wishes to evoke. With motion blur, a photographer is choosing to showcase the movements within their frame. If the camera's subject is moving fast or there's a longer exposure, you'll see more motion blur as a result. This blur effect usually occurs along the direction of the subject's movement, but if the camera is moving and the subject is still, then the background may become blurred in the same fashion.
Motion Blur in Video
Likewise, motion blur is readily apparent when shooting video. Most of the time, though, we don't even notice the effect because our eyes work the same way as a camera lens. To us, rapid movement through a still frame is supposed to look blurry. When we track a moving subject by panning the camera, however, the focal point looks sharper and the background blurs.
Motion Blur in 3D Animation/Games
Finally, digital animation and video games often simulate a motion blur effect to make the visuals seem less static. This is most obvious when quickly panning the camera around a 3D game world, or when CGI characters move rapidly in an animated scene. If digital designers didn't use any motion blur, the 3D visuals would seem choppier and less natural.
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