The collision happened on the A689 at Belle Vue Way at around 3.45pm on Sunday, June 5.
The driver of the private car, a 38 year-old man, had to be cut free by firefighters.
He was taken to hospital and later discharged.
11 of the most mispronounced place names in and around Hartlepool
Police appeal to find missing 40-year-old last seen in Billingham
Watch as a big explosion completely demolishes Sinter Plant
Police appeal to trace missing Hartlepool man Michael Rae
Plea to trace family of Hartlepool man in his sixties who died this week
A Cleveland Police spokesperson said: “Officers attending an urgent call just before 3.45pm yesterday, were travelling westbound on the A689 between Burn Road and Tomlinson Road when a vehicle collided with them.
"The 38 year-old male driver was cut from his vehicle, taken to James Cook University Hospital for treatment for minor injuries and discharged shortly afterwards.
“The man was arrested on suspicion of drug driving and other driving offences and he was bailed pending further enquiries.”
Two fire engines from Hartlepool Community Fire Station at Stranton nearby attended.
The ambulance service said it dispatched one paramedic crew.