The silver Kia Picanto car failed to stop for Humberside Police officers in Bridlington yesterday (Tuesday).
And, at around 10pm the same day, police were called over concerns about a car being driven dangerously on Columbus Ravine, Scarborough.
But after the car was traced to Cayton, it failed to stop after being instructed by North Yorkshire Police.
A spokesperson for the force said: "However, the vehicle rammed a police car and drove off from the area.
"Following a short pursuit, the vehicle was successfully brought to a stop in the Crossgates area of Scarborough.
"The male driver attempted to run off, but following a short foot pursuit, officers located and detained him."
A 19-year-old man, from Doncaster, has been arrested on suspicion of dangerous driving, taking a vehicle without consent, failing to stop and drink driving.