Police said two suspects stole a black VW Golf on Tuesday, July 12 this year.
And they sped along Leeds Road in Dewsbury, where it overtook a lorry before colliding with a white Ssangyong Korando.
The Korando was reversing into a driveway with the 35-year-old woman and her daughter inside.
The mum’s car was hit with such force that from a standstill it was sent into a 360 degree spin.
The woman was airlifted to hospital and required treatment for serious facial injuries. Her daughter was unharmed.
Both suspects fled the scene.
Police in Dewsbury have now released video footage taken from a dashcam in the lorry to try and find the drivers of the Golf
The images show two men they want to speak to in connection with the incident.
PC Gavin Smith, of Kirklees District Police, said: “This was a terrifying incident for the victim and her young daughter, one which has left the young girl terrified of getting into a car.
“Not only has the offending car been stolen, but it has been driven with such recklessness and speed it is miraculous that no one was more seriously injured.
“On colliding with the car, the suspects also left the scene with no regard for the occupants of the car they collided with.
“We are releasing these images today in the hope that members of the public will recognise the two men and will come forward to the police to help with our enquiries.
“Anyone with any information is asked to contact myself at Kirklees District Police, via 101 quoting reference 13160297787. Information can also be passed to Crimestoppers anonymously on 0800 555 111.”