Three people aged 24, 30 and 35, all from Leeds, have been arrested and remain in custody.
North Yorkshire Police said the officer was injured when approaching a suspicious vehicle in Bewerley Road, Harrogate, at 3.20pm on March 16.
A statement said: "The vehicle, a green Vauxhall Insignia, collided with the police officer, resulting in him falling over. The officer has been taken to hospital with a serious leg injury.
"The vehicle made off but was found a short time later.
"Anyone with any information is asked to contact the police. Dial 101, press 1 and ask to speak to the Force Control Room. Please quote reference number 12210078916 when passing on any information."