Considering the PS1 had two iterations of the Analog controller (The SCPH-1180 controller, and the more common DualShock controller), does the controller you're using have a rumble pack?
It's possible that when you're hitting a car, it's trying to run the rumble motor and something is happening to short the controller out. Can you try with another game that supports rumble and see if it also happens? Do you have another DualShock controller to test with as well?