No, sport mode is just as bad as any of the auto modes. Fine if you're satisfied with average-below par images, but if you want to get great images you'll need to take control yourself.
Using the panning technique will produce the most natural and stunning results, but it can be challenging. You should use continuous or servo focus which uses AI to predict where the subject will move to and keep it in focus. Combine this with the 55-200mm and shoot from a comforable distance. Select one AF point to the side where the car is moving towards.
Select Tv mode and choose 1/100 or 1/160 for slow moving cars, and 1/250-1/400 for faster cars. If you're shooting at a circuit, shoot at the apex of the turns, avoid the straights, especially with faster cars. Also engage continuous shooting/drive.
As a car is coming towards you half press the shutter when it's over the selected AF point, and start following the car with your camera, and fire off a burst when the car clips the apex.
Finally if there is any chance to ride in a car as a chase vehicle, jump at that opportunity, that allows you to use much slower shutter speeds, all you need to do is have the driver to match the speed of the target car. This is how all the car mags photograph their fast shots.