The police can give you a speeding ticket just by looking at you driving and making an estimate.

This may seem absurd and unfair, but it's the law, in the state of Ohio at least. Mark Jenney, a 27-year-old motorist in Akron, Ohio tried to appeal his speeding ticket. The officer had made the charge based on a "visual speed estimate", as opposed to using a radar, to judge how fast he was going. The 8th District Court of Appeals in Cleveland ruled that this is not a sufficient basis for writing a ticket.

The case went to the Ohio Supreme Court, who overturned the decision in a 5-1 ruling. According to the majority ruling:

"Rational triers of fact could find a police officer's testimony regarding his unaided visual estimation of a vehicle's speed, when supported by evidence that the officer is trained, certified by (the Ohio Peace Officer Training Academy) or a similar organization, and experienced in making such estimations, sufficient to establish beyond a reasonable doubt the defendant's speed. Independent verification of the vehicle's speed is not necessary to support a conviction for speeding."

(source)