You've no doubt seen in some sporting magazines, or even in the sporting section of the newspaper, photos of rally cars, or racing bikes in sharp, clear focus, yet their surroundings are all streaky and blurred.
This is brought about by an effect known as "panning" movement. To create this effect, you need to have your camera on a slow speed, small aperture, and the idea is to focus on the moving subject and as it passes, or reaches a predetermined point, only then press the shutter button, but whilst you are still moving with the subject.
It is advisable to have the camera mounted squarely on a sturdy tripod, with pan head fitted, for best results.
If you are using film, it is recommended that you use a slow film, 100 ISO, or set your camera's ISO rating to its lowest setting. Stop the lens down (select a small aperture, high number) of say, f11 or f16, with the slowest possible shutter speed for that setting, or if you can select shutter priority, try a range of speeds from between 1/60 second to 1/8 second and practice various shots.
It will take several goes before you get accustomed to this style of photography, but the results can be well worth the effort.
For that special shot, throw in some strong fill-flash, if you can, to enhance the subject's details. So even with that slow shutter speed, the flash will see that your subject is frozen in time.
Perhaps you could try out this technique on your children during their school sports meetings, or at football matches.
Incidentally, you are probably already aware, that you will now need permission to photograph your children or grand children at their sports meets, otherwise you can land yourself in big trouble. What is this world coming too, I ask you?
You can read more on this in "Your Rights and/or Restrictions".
Meanwhile, good luck with your panning.
Below are some examples of the panning technique and how your images should appear as a result: