A silver Honda CRV hit a a greyhound, called Silver, on Montpellier Hill in Harrogate at around 11.50pm on Thursday 11 April 2013.
The driver of the vehicle failed to stop at the scene, where the dog suffered a broken leg and a broken jaw.
Harrogate Police are appealing to any witnesses to the incident to come forward and help the investigation.
Traffic Constable Andy MacLeod, of Harrogate Roads Policing Group, said:
I am appealing to anyone who witnessed the collision to get in touch as soon as possible.
The dog sustained some nasty injuries in the incident and it is important that we trace the driver of the silver Honda.
If you were the driver of that vehicle, I urge you to do the right think and contact the police.
Anyone who can help officers with their enquiries is urged to contact North Yorkshire Police on 101 – select option 2 – and ask for Andy Macleod or Harrogate Roads Policing Group.
Alternatively, information can be sent via email to firstname.lastname@example.org
If you wish to remain anonymous, Crimestoppers can be contacted on 0800 555 111.
Please quote reference number 12130057516 when passing information about this incident.