Colin Fotheringham, 43, drove a VW Tiguan R-Line along the M9, then the M876 and finally the A9 towards Larbert.
A court heard the VW had "significant front end damage", and part of a plastic traffic cone was lodged underneath it.
Trailed by police, Fotheringham drove at 20 miles an hour along the M9, failing to stop when directed to so by the pursuing police car that was displaying flashing blue lights and "sounding its klaxons".
He then went as slow as ten miles an hour up the M876 in the direction of Dennyloanhead, driving onto the hard shoulder and swerving back onto both lane of the main carriageway, before turning onto the A9.
Prosecutor Rachel Wallace said the crew of a police van stationed at a roundabout on the A9 near Larbert saw him approaching.
He initially stopped at the rear of the police van, but then slowly accelerated again, colliding with the back of the van, which sustained "minor rear-end denting", and coming to a final complete halt.
Officers removed the keys from his ignition, and he was arrested.
The incident occurred around 02.20 am on January 14 last year.
Fotheringham, of Grangemouth, a former HGV driver, appeared for sentence at Falkirk Sheriff Court yesterday after pleading guilty last month to dangerous driving.
Solicitor Mark Fallon, defending, said: "He says he was under the influence of Valium.
"He accepts full responsibility for the offence and is embarrassed about it. His driving was horrendous, and the reason for it doesn't help."
Sheriff Derek Livingston told Fotheringham: "You're on the cusp of prison.
"There's no doubt that if someone chooses to go well under the speed limit on a motorway it can cause havoc.
"You were a real danger on the night in question, and the fact that you crashed into a police vehicle is indicative of that."
He banned Fotheringham from driving for 30 months, and sentenced him to 210 hours of unpaid work.