I was surprised by how much the figure of a person blurs as they walk across from left to right in front of the camera at a shutter speed of 1/60th second, so I double checked manual focus, bumped shutter speed up from recording 30p at 1/60 to 1/250, tried switching from 4K to HD and from 30fps to 60fps but the blurring of the moving figure was still a problem...

Several hours later I realized the problem was indeed shutter speed: clips filmed at 1/250th had less motion blur than 1/60 but were still not fast enough. This also surprised me as the person moving across the frame about 5 meters in front of the camera was walking at a normal walking pace and I had thought 1/250th would be more than enough. It wasn´t.

Trying again I found that I had to push the shutter speed up to a minimum of 1/800 second to get what I considered to be acceptable/pleasing blur (sufficiently sharp to be able to see details yet still enough blur to convey the feeling of motion. Even pushing the shutter speed up to max 1/2500 sec didn´t completely freeze the movement of the walker´s head.

Motion blur is essential for smooth looking motion in film and video. Removing it completely will produce a jerky look, some films like "Saving Private Ryan"use a higher shutter speed to intensify the explosions etc., however, it's about 1/250 at the most.

For normal work at 30fps, 1/60 is regarded as giving a natural look.

I was surprised by how much the figure of a person blurs as they walk across from left to right in front of the camera at a shutter speed of 1/60th second, so I double checked manual focus, bumped shutter speed up from recording 30p at 1/60 to 1/250, tried switching from 4K to HD and from 30fps to 60fps but the blurring of the moving figure was still a problem...

Several hours later I realized the problem was indeed shutter speed: clips filmed at 1/250th had less motion blur than 1/60 but were still not fast enough. This also surprised me as the person moving across the frame about 5 meters in front of the camera was walking at a normal walking pace and I had thought 1/250th would be more than enough. It wasn´t.

Trying again I found that I had to push the shutter speed up to a minimum of 1/800 second to get what I considered to be acceptable/pleasing blur (sufficiently sharp to be able to see details yet still enough blur to convey the feeling of motion. Even pushing the shutter speed up to max 1/2500 sec didn´t completely freeze the movement of the walker´s head.

What Brian said is absolutely correct- Motion blur helps the motion look smooth. Our eyes don't see the world at 30, or even 60, frames per second. When it comes to moving pictures, the blurring that results from the shutter speed is just one way our brains are tricked into thinking we are seeing smooth motion. Even though a still frame may look incredibly blurry, when watching at a normal playback speed, motion captured around 50 to 60hz will look natural, and the moving object will appear sharp as long as its in focus.

In regards to motion still not looking totally sharp even at very high shutter speeds, I suspect this is likely more of an issue of the codec you are using, and not your frame rate. A common way codecs compress a moving image is to take advantage of motion blur, compressing those areas with motion more than areas that are not changing. As a result, the blurring you are seeing at 1/2500 shutter speed may very well be introduced by your codec, not a result of the shutter speed not being fast enough.