It follows an accident in Holmfirth on Sunday morning in which a silver Honda S2000 convertible hit a telegraph pole on Dunford Road. It bought down power lines, cutting off the energy supply to nearby houses.
The 25-year-old driver of the Honda - which had its roof down - was airlifted to hospital with serious injuries.
He remains in hospital in a stable condition.
It is believed that a blue Ford Focus RS was at the scene and drove off at speed immediately after the collision.
Police Constable Cheryl Moore of the Safer Roads and Neighbourhoods Support Team West said; "I would appeal to anyone who was in the area on Sunday morning and who may have seen either vehicle prior to the collision or who may have information about the blue Ford Focus to come forward. I would also appeal to anyone who witnessed the collision itself to speak to police. Enquiries are continuing."