The black Renault was seen driving off from the scene following different collisions in the Ripon and Bedale areas on Wednesday afternoon.
North Yorkshire Police say one of the collisions, on the A1 near junction 50, was with a motorcyclist riding a Harley Davidson and left the cyclist seriously injured. The Renault driver did not stop at the scene.
A spokesman for the force said: "At around 12.35pm on the afternoon of Wednesday 17 July, the rider of a Harley Davidson motorcycle was knocked off his bike as he rode up the A1 northbound just north of junction 50 for Ripon/Baldersby.
"The vehicle involved, a black Renault Scenic, failed to stop at the scene and went on to be involved in four more collisions causing damage to vehicles in the Bedale, Carthorpe, Masham and West Tanfield areas."
Police confirmed that a man in his 30s from Wakefield was later arrested in Carthorpe on suspicion of multiple offences including dangerous driving, causing serious injury by dangerous driving and drink driving. He has been released while under investigation.
Detectives are now keen to hear from any witnesses who saw any of the collision, particularly those who may have dashcam footage or CCTV of the car as it passed by their property.
Anyone who can assist the investigation is asked to contact North Yorkshire Police on 101 and pass information for incident 12190129512.